Shell Method of solid generated by solid enclosed by $y=1/4x^3+2$ ,$ y=2-x$, and $x=2$ about $y$-axis

Question

The shell method is confusing to me, so can someone please help me out.

Find the volume of a solid generated by rotating the region enclosed by $y=\frac{1}{4} x^3 +2$ , $y=2-x$, and $x=2$ about the $y$-axis. Thanks.

Answer 1

Let's take a typical rectangle as in the picture, when it is rotated around $y$-axis we get a shell of volume $dV$ given by $dV=2\pi x \left[\frac{1}{4}x^3+2-(2-x)\right]dx$

The volume you are looking for can be found as $$V=\int_0^2 2\pi x \left[\frac{1}{4}x^3+2-(2-x)\right]dx$$