This is very simple but I can't figure out what I'm doing wrong.
Let's say I'm rotating $f(x)=x^2$ around the y-axis. I have limits on the x-axis of a=1 and b=2. I want the volume under the curve, so I use the shells method, which gives me:
$2\pi \int_a^b f(x)x\delta x$ = $\frac{15}{2}\pi$
Now let's say I do it another way. I'll find the volume of the hollow cylinder with outer radius $b$ and inner radius $a$ and height f(b). This is $12\pi$. Now I want to subtract what's above the curve rotated around the y-axis.
I rotate function f around the y-axis, this time getting the volume above the curve. I can do this by integrating $f^{-1}(y)$ with limits $f(a)$ and $f(b)$. That gives me:
$\pi\int_{f(a)}^{f(b)}(f^{-1}(y))^2dy = \pi\int_1^4(\sqrt(y))^2dy = \pi\int_1^4y dy=\frac{15}{2}\pi$.
I'm sure at this point I've already made my mistake since the last answer gave me my final answer before I was expecting it.
Where's my mistake?