Evaluating $\iiint_B (x^2+y^2+z^2)dV$ where $B$ is the ball of radius $5$ centered at the origin. 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
$$\int_B \rho^2\>dV=\int_0^5\rho^2\>4\pi\rho^2\>d\rho=2500\pi\ .$$
