Find the volume of the region inside both the sphere x^2+y^2+z^2=4 and the cylinder x^2+y^2=1.
I tried solving it using Fubinis theorem but the integral was too hard to solve?
Im just wondering how to use the change of variable theorem.
Thanks
Find the volume of the region inside both the sphere x^2+y^2+z^2=4 and the cylinder x^2+y^2=1.
I tried solving it using Fubinis theorem but the integral was too hard to solve?
Im just wondering how to use the change of variable theorem.
Thanks
First we can write this in term cylindrical coordinates this gives
$\displaystyle r^2+z^2=4 \iff z^2=4-r^2 \iff z=\sqrt{4-r^2}$ and $\displaystyle r^2=1$
This is only the top half of the region so the integral is
$\displaystyle 2\displaystyle \int_{0}^{2\pi}\int_{0}^{1}r\sqrt{4-r^2}drd\theta=2\left( \int_{0}^{1}r\sqrt{4-r^2}dr \right)\left( \int_{0}^{2\pi}d\theta\right)$