Evaluate $$\iiint_E z \,{\rm d}v,$$ where $E$ is the region between the spheres $x ^2 + y ^2 + z ^2 = 1$ and $x ^2 + y ^2 + z ^2 = 3$ in the first octant.
Using cylindrical coordinates I got $r$ bounded by $0 \leq r \leq \sqrt{2}$ and theta bounded by $0 \leq \theta \leq \pi/2$. $$\iiint_E 1 \,{\rm d}v$$
Let us use spherical coordinates
$$x=r\sin(\phi)\cos(\theta)$$
$$y=r\sin(\phi)\sin(\theta)$$
$$z=r\cos(\phi)$$
with $$1\leq r \leq \sqrt{3}$$ $$0\leq \theta \leq \frac{\pi}{2}$$
$$0\leq \phi \leq \frac{\pi}{2}$$
then $$I=\int_1^{\sqrt{3}}r^3dr\int_0^{\frac{\pi}{2}}d\theta\int_0^{\frac{\pi}{2}}\cos(\phi)\sin(\phi)d\phi$$
$$=\frac{\pi}{2}$$
Notice that we added the Jaccobian $$r^2\sin(\phi)$$