How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=1/x# and #2x+2y=5# rotated about the #y=1/2#?

Answer 1

#=pi(2 ln 2+21/8)=12.60# cubic units, nearly.

graph{ ( xy-1)(x+y-5/2) = 0 [-5.37, 5.37, -2.684, 2.684]}

The rectangular hyperbola H; #xy = 1# and the L: straight line
#x+y=5/2# meet at #(1/2. 2) and (2, 1/2)#.

The area is depicted by the graph.

The volume is

#pi int ((y_L-1/2)^2-(y_H-1/2)^2 )dx#, with x from #1/2# to 2
#=pi int ((5/2-x-1/2)^2-(1/x-1/2)^2) dx#,.with x from #1/2# to 2
#=pi int ((4-4x+x^2)-(1/4-1/(4x)+1/x^2) )dx#,
with x from #1/2# to 2
#=pi int (1/(4x)-1/x^2+15/4-4x+x2) dx#, with x from #1/2# to 2
#=pi[(1/4)(ln 2 -ln(1/2))+(1/2-2)+15/4(2-1/2)-2(2^2-1/2^2)-15+1/3(2^3-1/2^3)]#
#=pi[1/2 ln 2-3/2+45/8-15/2 -15+21]#
#=pi(2 ln 2+21/8)#, cubic units
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Answer 2

To use the shell method to set up and evaluate the integral for the volume of the solid generated by revolving the plane region (y=1/x) and (2x+2y=5) rotated about (y=1/2), follow these steps:

  1. Determine the limits of integration.

  2. Set up the integral using the shell method formula.

  3. Evaluate the integral.

  4. Limits of Integration:

    • To find the limits of integration, first, determine the points of intersection between the two curves.
      • Solve (y = \frac{1}{x}) and (2x + 2y = 5) simultaneously to find the points of intersection.
      • The intersection points will give the limits of integration for (x).
  5. Set Up Integral Using Shell Method:

    • The volume of the solid can be found using the shell method formula: [ V = 2\pi \int_{a}^{b} x \cdot h(x) , dx ] where (h(x)) is the height of the shell at each (x) and (a) and (b) are the limits of integration for (x).

    • For each (x), the height (h(x)) is the difference between the two curves. Since we are revolving around (y=1/2), the height will be the difference between the (y)-value of the point on the curve and (1/2).

  6. Evaluate Integral:

    • After setting up the integral, integrate with respect to (x) to find the volume of the solid.

By following these steps, you can use the shell method to set up and evaluate the integral for the given problem.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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