We place the centre of the cask at the origin, and assume that the cask is symmetrical about its mid-plane. 

Consider the parabola $y=40-kx^2$. At $x=50$ we want $y=30$. So $k=\frac{10}{2500}=\frac{1}{250}$.  

We will rotate this parabola about the $x$-axis. The resulting solid is the cask.

It is useful to integrate from $x=0$ to $x=50$, and double. By the usual formula, the volume, in cubic cm, is
$$2\int_0^{50}\pi\left(40-\frac{x^2}{250}\right)^2\,dx.$$

Remark: Kepler once wrote a little book about how to compute the volumes of Austrian wine casks. The book did not sell well, and Kepler had to go back to casting horoscopes and theorizing about the motions of the planets. 

Find volume of cask
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