Find the total mass of the solid defined by the inequalities $x^2 + y^2 + z^2 \ge 1, \hspace{.1cm} x \ge 0,\hspace{.1cm} y \ge 0$ with mass density $z^2$.
I know I have to use triple integrals to evaluate but I am confused as to what limits I should use.
The volume is a sphere with radius $1$, centered at the origin with a restriction on the azimuthal angle $\phi$. Since $x\ge0$ and $y\ge 0$ We restrict the azimuthal angle so that $0\le \phi \le \pi/2$.
The polar angle $\theta$ spans over its complete domain with $0\le \theta\le \pi$ while the radial variable extends from $0$ to $1$.
Recalling that the Jacobian $J$ of Transformation from Cartesian to Spherical coordinates is $J=r^2\sin \theta$, we have
$$\begin{align} M&=\int_{0}^{\pi/2}\int_{0}^{\pi}\int_{0}^{1}(r^2\cos^2\theta)r^2\sin\theta \,dr\,d\theta\,d\phi\\\\ &=\frac{\pi}{2}\times \frac23\times\frac15\\\\ &=\frac{\pi}{15} \end{align}$$