Disk method problem

[Funkychemist]

Ok got one more I'm stumped on.

Find the capacity of a wine barrel with the shape of a solid that is obtained by revolving the region bounded by the graphs of $\displaystyle x = R - Ky^2$, $\displaystyle x = 0$, $\displaystyle y = \frac{-h}{2}$, and $\displaystyle y = \frac{h}{2}$ about the y-axis.

So far I've set up an integral from [$\displaystyle \frac{-h}{2}$,$\displaystyle \frac{h}{2}$] of $\displaystyle x = (R-Ky^2)^2$ all multiplied by pi, however, I get stuck when trying to integrate it.

 

[wjm11]

x = R – Ky^2

Just expand it before integrating:

x^2 = (R – Ky^2)^2 = R^2 – 2RKy^2 + (K^2)(y^4)