Let $T$ be a triangle in the $yz$-plane, with vertices at $(0,0,0)$, $(0,1,1)$and $(0,−1,1)$. Define $C$ to be the cone resulting from rotating $T$ around the $z$-axis.
Let $S$ be the unit sphere centered at $(0,0,0)$; that is, let $$ S=\{(x,y,z)\in\mathbb{R}^3\colon x^2+y^2+z^2\le 1\}. $$
Finally define $R\colon = C\cap S$, that is, $R$ is the intersection of $C$ and $S$.
One could compute the volume of $R$ via:
(Find the limits of integration in cartesian coordinates, cylindrical coordinates, and spherical coordinates).
I understand what each coordinate system is, but I'm really having trouble coming up with the limits. When someone explains a solution to me, I understand the logic behind it, but could someone help me and try to explain how to find the limits given this information? A solution + explanation would be greatly appreciated. I would like to know how to approach problems like this in general.
Let's try to understand the region $R$ first. The generated cone is not contained inside the unit sphere since the point $(0,1,1)$ and $(0,-1,1)$ is outside the sphere. Since the cone is opening upward in the $z$-direction, the region $R$ can be seen as follows: it's the region from the curved surface of the cone extended all the way until you hit the surface of the upper hemisphere.
It is natural to express the region in terms or spherical coordinates. Let $\theta$ be the azimuthal angle and $\phi$ the polar angle. The range of $r$ and $\theta$ should be clear: $$ 0\le r\le 1, \ \ 0\le \theta\le 2\pi. $$ The polar angle $\phi$ ranges from 0 to $\pi/4$, where $\pi/4$ is obtained by solving for the angle measured from the $z$-axis to the side of the cone (or in terms of the triangle, the angle from the $z$-axis to the side of the triangle.) Hence, the region $R$ in spherical coordinates is $$ R = \left\{(r,\theta,\phi)\colon 0\le r\le 1, 0\le \theta\le 2\pi, 0\le \phi\le \pi/4\right\}. $$