I am taking calculus 3 and I have problems understanding how to change from rectangular coordinates to spherial ones (integration).
For example, I have this problem:
Find the volume of the solid $T$ enclosed by the surface:
$$(x^2 + y^2 +z^2)^2 = 2z(x^2 + y^2)\;.$$
How I change to spherical ones? Can someone resolve this one explaining it step by step??
Also if someone know a place in the web to learn this it will be appreciated! Thanks!!
You can look at Wikipedia. You have $x=r \sin \theta \cos \phi, y=r \sin \theta \sin \phi, z= r\cos \theta$ The volume element is $r^2 \sin \theta dr\;d\theta\; d\phi$ Your equation becomes $r^4=2r^3 \cos \theta \sin^2 \theta$ by plugging in the expressions.