Evaluate the following triple integral as a repeated integral using the appropriate coordinate system:

$\int \int {\int }_S{x}^2.\mathrm{dV}$ where $S$ is the unit ball $\{\rho ,\varphi ,\theta ):\rho <1\}$.

Any help would be great.

In a multiple integral, if you change variables from (x,y,z) to (u,v,w), then



where



and you multiply the integrand by the absolute value of this determinant.

$x=\rho \sin \varphi \cos \theta$

$y=\rho \sin \varphi \sin \theta$

$z=\rho \cos \varphi$.

Think about what limits you would need to cover the entire sphere of radius $\rho$ after making this transformation. Be careful that each unit volume is only covered once.