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#, #y=x^2#, and #x=0# rotated about the line #y=2#?

Answer 1

#(28pi)/15# cubic units

Since we are revolving around a horizontal line using the method of shells we will integrate with respect to #y#.
We are bounded by the #y# axis, the horizontal line #y=1#, and the function #y=x^2#
Solve #y=x^2# for #x#
#x=sqrt(y)#
We are in quadrant #I# so we do not have to worry about the negative square root.
Our representative cylinder height is our function #sqrt(y)#
Our representative radius is #2-y# over the interval #0<=y<=1#

The integral for the volume is

#2piint_0^1(2-y)(y^(1/2))dy#
#2piint_0^1 2y^(1/2)-y^(3/2)dy#

Integrating we get

#2pi[4/3y^(3/2)-2/5y^(5/2)]#

Evaluating we get

#2pi[4/3-2/5-0]#
#2pi[20/15-6/15]=2pi[14/15]=(28pi)/15# cubic units
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Answer 2

To 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), (y=x^2), and (x=0) rotated about the line (y=2), we first need to find the limits of integration and the expression for the radius and height of the cylindrical shells.

The limits of integration will be determined by the intersection points of the curves (y=1) and (y=x^2), which occur at (x=1) and (x=-1).

The expression for the radius of the cylindrical shells will be the distance from the axis of rotation ((y=2)) to the curve (y=x^2), which is (2 - x^2).

The expression for the height of the cylindrical shells will be the difference between the upper and lower (y)-values of the region, which is (1 - x^2).

Thus, the integral to find the volume of the solid is:

[ V = 2\pi \int_{-1}^{1} (2 - x^2)(1 - x^2) , dx ]

Solving this integral will give the volume of the solid generated by the rotation.

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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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