Question

Let E be the solid in the first octant bounded on the sides by the xz plane and the plane y=x, above by the sphere x^2+y^2+z^2=16, and below by the cone z = sqrt (3x^2+3y^2). Express the volume of E as triple iterated integrals in spherical and cylindrical coordinates.

Answer 1

Hi. My answer in cylindrical coordinates are as follows:
\[\int\limits_{0}^{\pi/4}\int\limits_{0}^{4}\int\limits_{\sqrt(3)r}^{\sqrt(16-r^2)} r dz dr d \theta \]

Answer 2

According to my answer key the limits of integration for the radius us 0 to 2 and not 0 to 4. Can someone help me understand please? :)