The aim in choosing the limits of integration (and the coordinates you use) is to make sure each point in the space of interest is counted exactly once (well, technically you can double-count or not count some points, provided there are few enough of them). This means that to find the limits you need to consider where your coordinates change from points inside the space you want to outside (or start double-counting, or become meaningless eg negative radius).
For example, to over the circle $x^2+y^2=4$, you want radius between $0$ and $2$, and angle between $0$ and $2\pi$. If you wanted the annulus between the circles $x^2+y^2=4$ and $x^2+y^2=16$ then you would want radius between $2$ and $4$.
Without working out the details, I assume in this question you will need to change the shape you are integrating over as the $z$ coordinate changes, so sometimes you are interested in the cylinder and sometimes in the ellipsoid.