Thread: Multiple Integrals (Volume + Centroid)

A) Find the volume bounded by sphere rho=rt(6) and the paraboloid
z=x^2 + y^2
B) Locate the centroid of this region

Any help, tips, work, or similar problems would be greatly appreciated. =)

I start you off. The upper surface and the lower surface . When they intersect the region over we are integrating is the circle . Now use polar change of variable.

Originally Posted by ThePerfectHacker

I start you off. The upper surface and the lower surface . When they intersect the region over we are integrating is the circle . Now use polar change of variable.

When you say polar change of variable you mean things like
x = rcos theta
y = rsin theta
z = z
right?
and then i would take the triple integral of that?
I always have trouble setting up the problem. =(
Like setting the integral ranges.

Originally Posted by numberonenacho

A) Find the volume bounded by sphere rho=rt(6) and the paraboloid
z=x^2 + y^2
B) Locate the centroid of this region

The upper surface (sphere) is given by .
Thus, the volume is,
where is the disk of radius .
Using polar change of variable,

When I tried to evaluate the integral, I can do the R part of it, but theres no place for me to put the theta value. Shouldn't there be two different variables in a double integration problem like this?

Is that ?.

If so, try:

hmm okay. So when I evaluate it




how do I do the second part of the integral. That rdr complicates everything.