# 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 about the y-axis given #y=16x-x^2#, x=0, and y=64?

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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 about the y-axis, given ( y = 16x - x^2 ), ( x = 0 ), and ( y = 64 ), first express the function ( y = 16x - x^2 ) in terms of ( x ). Then, consider a vertical strip of thickness ( dx ) at a distance ( x ) from the y-axis.

The height of the cylinder formed by revolving this strip is the difference between the upper and lower y-values, which is ( 64 - (16x - x^2) ). The circumference of the cylinder is ( 2\pi x ), and the thickness of the shell is ( dx ).

Thus, the volume of the shell is ( 2\pi x \times (64 - (16x - x^2)) \times dx ). Integrate this expression from ( x = 0 ) to the x-coordinate where ( y = 64 ) to find the total volume of the solid.

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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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