# Iterated Intergal

• Mar 29th 2010, 09:03 AM
guess
Iterated Intergal
Find the volume of the solid inside the cylinder x^2+y^2 = 2y and bounded above and below by the sphere x^2 + y^2 + z^2 = 4.

Thanks for visiting through my problems. I need to get the solution but i do not have any idea on how to solve it. Please help. Thanks
• Mar 29th 2010, 11:59 AM
HallsofIvy
Quote:

Originally Posted by guess
Find the volume of the solid inside the cylinder x^2+y^2 = 2y and bounded above and below by the sphere x^2 + y^2 + z^2 = 4.

Thanks for visiting through my problems. I need to get the solution but i do not have any idea on how to solve it. Please help. Thanks

By completing the square, that cylinder is $\displaystyle x^2+ (y- 1)^2= 1$ which has axis (0, 1, z) which is slightly of center from the sphere. I think the first thing I would do is "translate" the origin to (0, 1, 0)- replace y by y'= y- 1 so that y= y'+ 1 and the equation of the cylinder is $\displaystyle x^2+ y'^2= 1$ and the equation of the sphere is $\displaystyle x^2+ (y'+ 1)^2+ z^2= 4$ or $\displaystyle x^2+ y'^2+ z^2+ 2y= 3$.

Now the cylinder has its axis at (0, 0, z) and you can use cylindrical coordinates: z will run from the part of the sphere at the bottom, $\displaystyle z= -\sqrt{3- x^2- y^2- 2y}= -\sqrt{3- r^2- 2rsin(\theta)}$, to the part at the top, $\displaystyle z= \sqrt{3- x^2- y^2- 2y= \sqrt{3- r^2- 2rsin(\theta)}$ while r and $\displaystyle \theta$ range from 0 to 1 and 0 to $\displaystyle 2\pi$. The volume is given by
$\displaystyle \int_{\theta= 0}^{2\pi}\int_{r= 0}^1\int_{z=-\sqrt{3- r^2- 2rsin(\theta)}}^{\sqrt{3- r^2- 2rsin(\theta)}} r dz dr d\theta$
• Mar 29th 2010, 02:12 PM
shawsend
Hi. Here's an alternate way just for fun. Do the complete the square thing, then convert $\displaystyle x^2+(y-1)^2=1$ to polar coordinates to obtain $\displaystyle r=2\sin(t)$, and the sphere into cylindrical coordinates $\displaystyle z^2=\sqrt{4-r^2}$, then the integral is:

$\displaystyle 4\int_0^{\pi/2}\int_0^{2\sin(t)}\int_0^{\sqrt{4-r^2}} rdzdrdt$ which is the same answer as HallsofIvy above, about 9.6 which is in the ball park since the volume of the whole cylinder of length 4 is $\displaystyle 4\pi\approx 12.6$