I am taking a Calculus course currently and am stuck on the last question of my assignment.
Find the volume of the region inside the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ and above the plane $z = b - y$.
I've tried solving the ellipsoid equation to get bounds on the triple integral, but it seems intractably hard, and I tried the change of coordinates $\beta = z + y$, $\gamma - (x+y)$, and $\lambda = x$. The Jacobian of this transformation has determinant 1 and should rotate $\mathbb{R}^3$ so that the plane $z=b-y$ is our "new" xy-plane. In any case, I haven't been able to figure it out using this approach either. Any help would be greatly appreciated!