# Explain this limit of integration for radius in polar coordinates.

Use polar coordinates to find the volume of the given solid:

Inside both the cylinder $x^2 + y^2 = 4$ and the ellipsoid $4x^2 + 4y^2 + z^2 = 64$

The limit of integration for the radius goes from 0 to 2 because the problem asks for the solid inside the cylinder? If it asked for the solid between the cylinder and ellipsoid it would be from 2 to 4?

• Provided that polar coordinates at XY plane, radius of your cylinder is 2, and your ellipsoid is 4. z coordinate will depend on r. – 0x2207 Dec 9 '12 at 15:32

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.