The question asks to use spherical coords. My answer is coming out wrong and symbolab is saying I'm evaluating the integrals correctly so my set up must be wrong.
Since $\rho$ is the distance from the origin to a point on it, and it's a sphere, I got $0 \le \rho \le 5$
Since it's a sphere I did $\theta$ from $0$ to $2\pi$. And then for $\phi$ I have from $0$ to $\pi$.
From an example problem, $x^2+y^2+z^2=\rho^2$
Thus
$$\int^\pi_0\int^{2\pi}_0\int^5_0 [( \rho^2) \rho^2 \sin(\phi)]\,d\rho \,d\theta \,d\phi$$
The answer is $\frac{312,500\pi}{7}$ and I'm getting $\frac{-1250}{\pi}$.
Partition your ball $B$ into spherical shells of radius $\rho$ $(0\leq\rho\leq5)$ and thickness $d\rho$. The volume of such a shell is $dV=4\pi \rho^2\,d\rho$. It follows that $$\int_B \rho^2\>dV=\int_0^5\rho^2\>4\pi\rho^2\>d\rho=2500\pi\ .$$