Consider a cone centered about the positive z axis with its vertex at origin,a $90^{\circ}$ angle at its vertex,topped by a sphere of radius $6$.Compute the volume of region bounded by sphere and cone.
My problem:I need help about the limits of integration in spherical coordinates.
Thanks.
Best Answer
A picture always helps. The green region is the one you are interested in (assuming sphere centered at origin)
The limits of integration for spherical coordinates as you rightly noted in comments in the other post are $0\leq r \leq 6$ and $0\leq \theta \leq 2\pi$
For finding the limits for $\phi$, refer to the picture and note that the problem specified that the angle of the black triangle at the vertex is $90^\circ = \pi/2$. Consider how much of an angle from the red line you will need to go down (hint: the red line bisects the angle) to find the largest $\phi$ can be.