How do you find the surface area of the solid obtained by rotating about the #x#axis the region bounded by #y=x^3/6+1/(2x)# on the interval #1/2<=x<=1# ?
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To find the surface area of the solid obtained by rotating the region bounded by y = x^3To find the surface area of the solid obtained by rotating the given region about the xaxis:
1To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6To find the surface area of the solid obtained by rotating the given region about the xaxis:
1.To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 +To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representingTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2xTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve:To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) onTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: yTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = xTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 +To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 aboutTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2xTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x). 2.To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxisTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

FindTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:
To find the surface area of the solid obtained by rotating the given region about the xaxis:
 Determine the function representing the curve: y = x^3/6 + 1/(2x).
 Find theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:
1.To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivativeTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

DetermineTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of yTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y withTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula forTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect toTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to xTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface areaTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x. 3To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolutionTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

UseTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution,To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which isTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula forTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is givenTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surfaceTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integralTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area ofTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotatedTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πyTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SATo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA =To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dyTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dxTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a,To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, bTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dxTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx
To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx
2To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πyTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

FindTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dyTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dxTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivativeTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dxTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of yTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y =To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2)To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = xTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dxTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.
To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.
4To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.
4.To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

SubstituteTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 +To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the valuesTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values ofTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of yTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2xTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y andTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x)To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dyTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) withTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dxTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respectTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx intoTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect toTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to xTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formulaTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.
To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula. To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.
3To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula. 5To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

PlugTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula. 5.To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

IntegrateTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression forTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for yTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression withTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y andTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respectTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dyTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect toTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formulaTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given intervalTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.
4To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.
4.To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

IntTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

IntegrateTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2,To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resultingTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expressionTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression overTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over theTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following theseTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the givenTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these stepsTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps willTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give youTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface areaTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ xTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of the solidTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ 1.
To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of the solid obtained byTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ 1.
5.To find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of the solid obtained by rotating the givenTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ 1.

CalculateTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of the solid obtained by rotating the given regionTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ 1.

Calculate the definiteTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of the solid obtained by rotating the given region about theTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ 1.

Calculate the definite integral toTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of the solid obtained by rotating the given region about the xTo find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ 1.

Calculate the definite integral to findTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of the solid obtained by rotating the given region about the xaxis.To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ 1.

Calculate the definite integral to find the surfaceTo find the surface area of the solid obtained by rotating the given region about the xaxis:

Determine the function representing the curve: y = x^3/6 + 1/(2x).

Find the derivative of y with respect to x.

Use the formula for the surface area of a curve rotated about the xaxis:
SA = ∫[a, b] 2πy √(1 + (dy/dx)^2) dx.

Substitute the values of y and dy/dx into the formula.

Integrate the expression with respect to x over the given interval [1/2, 1].
Following these steps will give you the surface area of the solid obtained by rotating the given region about the xaxis.To find the surface area of the solid obtained by rotating the region bounded by y = x^3/6 + 1/(2x) on the interval 1/2 ≤ x ≤ 1 about the xaxis:

Determine the formula for the surface area of revolution, which is given by the integral: ∫ 2πy √(1 + (dy/dx)^2) dx

Find dy/dx by taking the derivative of y = x^3/6 + 1/(2x) with respect to x.

Plug the expression for y and dy/dx into the formula.

Integrate the resulting expression over the given interval, 1/2 ≤ x ≤ 1.

Calculate the definite integral to find the surface area.
The surface area of the solid can be found using the formula and the steps outlined above.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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