Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
$y=32-x^2, \ y=x^2$ about the line $x=4$
My confusion is that what will be the radius 'x' of cylinder shells which we have to put in the integral.
Best Answer
Let $-4\le x\le 4$. Draw a thin vertical strip of width "$dx$" at $x$. For the picture, let $x$ for example be $1.5$.
Rotate this thin strip about the line $x=4$. We get a cylindrical shell. This shell has height $(32-x^2)-x^2$. Its distance from the line $x=4$ is $4-x$. That is the radius of the cylindrical shell. It follows that the volume is equal to $$\int_{-4}^4 2\pi(4-x)(32-2x^2)\,dx.$$