# Finding Volume Using Cylindrical Shells

It says use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis y=x^3 y=8 about the axis x=3 I drew the graph, reflected it about the x=3 line, and drew a cylinder. `I figured that the radius is just r=3-x and the height would just be h=x^3-0 (since the lowest y value is zero), and plugged these into the integral of 2(pi)(r)(h) from 0 to 2. However, I got the wrong answer (correct answer should be 264pi/5).

I have a feeling that my height may be wrong but I'm not sure why.

Thank you!

• I think you left out some information in the problem above. You say the bounds are $y=x^3$ and $y = 8$ but is it also bounded by the $x$ and $y$ axes? Commented Dec 7, 2016 at 21:08

Assuming the $x$ and $y$ intercepts are bounds in this problem (because you didn't mention it) the two equations intersect at $x = 0$ and $x = 2$, so those are the bounds of integration. By rotating around the line $x = 3$, the radius of the shell is $3-x$ and the height of the shell is $8-x^3$
Using the shell method, the integral would be set up like so: $$2\pi\int^2_0(3 - x)(8 - x^3) dx$$