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.
$$\begin{array}{rcl}<br /> x & = & r\sin\theta\cos\varphi \\<br /> y & = & r\sin\theta\sin\varphi \\<br /> z & = & r\cos\theta<br /> \end{array}$$

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