Evaluate:
$$\iiint_V (x^2+y^2)\:\mathrm{d}x\:\mathrm{d}y\:\mathrm{d}z,$$
where $V$ is the region in the positive octant bounded by the sphere $\|\vec{r}\|=a$.
I am unsure how to get the limits of integration here.
I believe I need to use cylindrical coordinates
$x = r\cos \theta$
$y = r\sin \theta$
$z=z$
$$\iiint_V f(x,y,z)\:\mathrm{d}x\:\mathrm{d}y\:\mathrm{d}z = \iiint_V f(r\cos \theta, r\sin \theta, z)r \:\mathrm{d}z\:\mathrm{d}r\:\mathrm{d}\theta = \iiint_V r^3 \:\mathrm{d}z\:\mathrm{d}r\:\mathrm{d}\theta,$$
But because the bound is only $\|\vec{r}\|=a$.. i'm unsure what to do now.
Or am i completely wrong :L?
The boundary is a piece of a sphere so use spherical coordinates:
$$dV = r^2 \, \sin{\theta} \, dr \, d\theta \, d\phi$$
$$x^2+y^2=r^2 \sin^2{\theta}$$
Since we are in the first octant, the integral is
$$\int_0^a dr \, r^4 \, \int_0^{\pi/2} d\theta \, \sin^3{\theta} \, \int_0^{\pi/2} d\phi$$