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?

Question

Evelyn Agard · Accepted Answer

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Christopher Allers · Answer

undefined 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.