I'm struggling to find my mistake in the following problem:
Let $S$ be a sphere of radius $R$ with center at the origin. Let $f(x,y,z) = 3z^2$. Find the integral of $f$ over the volume of the sphere.
My solution is below. Where is my mistake?
The volume of the sphere can be parameterized using spherical coordinates $(\theta, \phi, r)$ with $$\theta \in [0, 2 \pi] \\ \phi \in [- \pi, \pi] \\ r \in [0, R] \\ z = r \sin \phi$$ giving $$\Phi = 3 \int_0^{2\pi} \int_{-\pi}^{\pi} \int_0^R r^2 \sin^2 \phi \, dr \, d\phi \,d \theta.$$
Since the integrand is not a function of $\theta$, this simplifies to $$\begin{align} \Phi &= 6 \pi \int_{-\pi}^{\pi} \int_0^R r^2 \sin^2 \phi \, dr \, d\phi \\ &= 2 \pi \int_{-\pi}^{\pi}[ r^3 \Big|^{R}_0] \sin^2 \phi \, d\phi \\ &= 2 \pi^2 R^3.\end{align}$$
Yet the source states the answer is $\frac {4 \pi R^5} 5$. What is my mistake?
Update
From the comments, it appears I'm missing something basic. That is, this was not a careless mistake, but rather a gap in my understanding of how to do these types of integrals.
It seems I assumed the area of each small region is constant, namely $dr \, d\theta \, d \phi$. But this is clearly not the case, since, for example, a constant change in $\theta$ will cover much more volume when $r$ is great than when its small.
This may be what Guliano meant by "you forgot the volume element in spherical coordinates" and Cunyi Nan meant by "You forgot to multiply the Jacobian". I would request, if possible, elaboration.
Use spherical coordinates: $x=r\sin\phi\cos\theta,y=r\sin\phi\sin\theta,z=r\cos\phi$. Here $0\leq\theta\leq2\pi,0\leq\phi\leq\pi,0\leq r\leq R$.
The Jacobian of it is $\frac{\partial(x,y,z)}{\partial(r,\theta,\phi)}=r^2\sin\phi$.
So the integral is $$I=\iiint_{V}3z^2\mathrm{d}V=3\int_{0}^{2\pi}\mathrm{d}\theta\int_{0}^{\pi}\mathrm{d}\phi\int_{0}^{R}(r^2\sin\phi)\cdot(r^2\cos^2\phi)\mathrm{d}r=\frac{4\pi R^5}{5}$$