Find the volume of the solid region inside the sphere $x^2+y^2+z^2=6$ and above the paraboloid $z=x^2+y^2.$
I set both equations equal to each other and obtained $z=2$ and $z=-3$. Since clearly $z>0,$ that means I only consider $z=2.$ Subbing this into both equations gives $x^2+y^2=2$.
I sketch this circle and obtain $0 \leq \theta \leq 2 \pi$ and $0 \leq r \leq \sqrt2.$ Also, clearly $r^2 \leq z \leq \sqrt{6-r^2}.$
Now I want to convert into spherical coordinates. I don't know how to find the limits for $\phi$. I think the lower limit is $0$ but I am not sure about the upper limit. I think the upper limit for $p$ is $6$. Not sure how to find the rest.