Question

how do you use spherical polar coordinates to evaluate the triple integral over B of z^9 dV

where B is the hemisphere 0<= z <= (1-x^2-y^2)^1/2???

Answer 1

\[x=r\sin\theta\cos\phi\quad,\quad y=r\sin\theta\sin\phi\quad,\quad\quad z=r\cos\theta\quad,\quad dV=r^2\sin\theta drd\theta d\phi\]\[0\leq r\leq 1\quad,\quad 0\leq\theta\leq \pi/2\quad,\quad 0\leq\phi\leq 2\pi\]