What is the centroid of a region D lying above a sphere and below a paraboloid?

[MozAngeles]

Homework Statement

Find the centroid of the region D, lying above the sphere x²+y²+z² and below the paraboloid z=4-x²-y².

Homework Equations

The Attempt at a Solution

I decided it might be easier to change it to polar coordinates soo I did
M=∫∫∫ dzrdrdθ
where z goes from sqrt(6-r²) to 4-r²
θ goes from 0 to 2∏
then i thought the r would be where the two functions intersect, so i set sqrt(6-r²)=4-r² solving for r i got my solutions came out to be sqrt2 and sqrt5, i don't know which one to use because i thought r would go from 0 to either sqrt2 or sqrt5

if some one could help me out i would really appreciate it.. thanks
solving for r i got

 

[HallsofIvy]

Look at a graph. $r^2+ z^2= 6$ (you forgot the "6" in your origina equation but it can be deduced from what you wrote later) and $z= 4- r^2$ intersect twice. For $r= \sqrt{2}$, $z= 4- 2= 2$ and for $r= \sqrt{5}$, $z= 4- 5= -1$. Because the problem says "above the sphere and below the paraoloid, you want to use the higher one, $r= \sqrt{2}$, z= 2.