Centroid of a 3D Region using Triple Integral

[jj2443]

Homework Statement

Compute the centroid of the region defined by x^{2} + y^{2} + z^{2} \leq k^{2} and x \geq 0 with \delta(x,y,z) = 1.

Homework Equations

\overline{x}=\frac{1}{m}\int\int\int x \delta(x,y,z) dV

\overline{y}=\frac{1}{m}\int\int\int y \delta(x,y,z) dV

\overline{z}=\frac{1}{m}\int\int\int z \delta(x,y,z) dV

The Attempt at a Solution

I understand that I need to integrate each of the above equations to get the x,y,z coordinates of the centroid, but how do I determine the bounds of integration?

Any help would be much appreciated! Thanks!

 

[hunt_mat]

I would use spherical polar co-ordinates.
<br /> \begin{array}{rcl}<br /> x &amp; = &amp; r\sin\theta\cos\varphi \\<br /> y &amp; = &amp; r\sin\theta\sin\varphi \\<br /> z &amp; = &amp; r\cos\theta<br /> \end{array}<br />

with dv=r^{2}\sin\theta drd\theta d\varphi