Use cylindrical coordinates to set up the integral of f(x, y, z) over the region outside the cone z^2=x^2+y^2 and inside x^2+y^2+z^2=9. Please don't solve it for me, I'm just having trouble finding the limits of integration.

Question

Use cylindrical coordinates to set up the integral of f(x, y, z) over the region outside the cone z^2=x^2+y^2 and inside x^2+y^2+z^2=9.  Please don't solve it for me, I'm just having trouble finding the limits of integration.

blockcolder · Answer

Use the conversion equations from Cartesian to cylindrical and find out where the cone and the sphere intersect. :D

anonymous · Answer

Um... they intersect at x^2+y^2=9-x^2-y^2.  So r=3/rad(2).  Still confused.  This is one of the first problems I've done of this type.  Still confused on the limits.

blockcolder · Answer

I'd divide the region into two parts but it would be ugly. :(

anonymous · Answer

Help would still be appreciated.

anonymous · Answer

$$
\int
   _0^{\frac{3}{\sqrt{2}}}\int
   _r^{\sqrt{9-r^2}}rdzdr = 9-\frac{9}{\sqrt{2}}
$$

blockcolder · Answer

Aren't you lacking a d(theta)?

anonymous · Answer

Multiply by 2 pi the above answer.

blockcolder · Answer

Oh yeah, My bad. =))

anonymous · Answer

$$
\int_0^{2 \pi} 
\int _0^{\frac{3}{\sqrt{2}}}\int
   _r^{\sqrt{9-r^2}}rdzdrd\theta = 2\pi \left(9-\frac{9}{\sqrt{2}}\right)
$$

anonymous · Answer

You can practice similar problems and more on http://moltest.missouri.edu/mucgi-bin/calculus.cgi Chose CalcIII (Multiple Integrals)