Question

Set up the integral that uses the method of cylindrical shells to find the volume v of the solid obtained by rotating the region bounded by the given curves about the specified line. 64y = x3, y = 0, x = 8 about the line y = 8

Answer 1

Given curves:

$64y=x^3, y=0, x=8$

The region is rotated about the line $y=8$

Shell method formula : $\int\limits^b_a {2\pi rh } \, dy$

Since the region is rotated about the line y= 8, we solve the given curve for x.

Take cube root on both sides.

The radius of the given region is $y-8$ and the height is $x=4\sqrt[3]{y}$.

The integral to find the volume is :  $\int_{8}^{16}2\pi (y-8)(4\sqrt[3]{y})dy$