# Math Help - Change of variable problem

1. ## Change of variable problem

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

2. Originally Posted by farmeruser1
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

$r^2+z^2=4 \iff z^2=4-r^2 \iff z=\sqrt{4-r^2}$ and $r^2=1$

This is only the top half of the region so the integral is

$2\displaystyle \int_{0}^{2\pi}\int_{0}^{1}r\sqrt{4-r^2}drd\theta=2\left( \int_{0}^{1}r\sqrt{4-r^2}dr \right)\left( \int_{0}^{2\pi}d\theta\right)$

3. So I had the right integral,
but can you just explain to me why the boundary for theta is 2pie and 0? and also why is there a 2 in front of the whole integral? thanks