Volume of a wine barrel

Let $k=h/2$, and put the origin in the middle, where symmetry asks it to be.

Then the equation of the upper parabola is $$y=b-\frac{b-a}{k^2}x^2.$$ The integral of $\pi y^2\,dx$ from $0$ to $k$ is $$\pi k\left(b^2-\frac{2}{3}(b-a)b+\frac{1}{5}(b-a)^2\right).$$ This simplifies to $$\frac{\pi k}{15}(3a^2+4ab+8b^2)$$ Replace $k$ by $h/2$ and multiply by $2$.