So I'm gonna replace my stuff with spherical coordinates with the following equations:
$$x^2+y^2+z^2\:=\:\rho \:cos^2\left(\theta \right)+sin^2\left(\theta \right)sin^2\left(\phi \right)+cos^2\left(\phi \right)$$ And when I simplify:
$$\rho ^2=x^2+y^2+z^2,\:\rho ^2=25,\:\rho \:=5$$
Then since it's a ball I think: $$0\le \theta \le 2\pi $$
And then $$z = \frac{5}{2}$$
so:
$$(5)cos{\phi} = \frac{5}{2} , \phi = \frac{\pi}{3}$$
And so my integral is:
$$\int _{0\:}^{2\pi \:}\:\int _{0\:}^{\frac{\pi }{3}\:}\:\int _{\frac{1}{cos\left(\phi \right)}}^{\:5}\:\:\:\rho \:\:cos^2\left(\theta \:\right)+sin^2\left(\theta \:\right)sin^2\left(\phi \:\right)+cos^2\left(\phi \:\right)d\rho \:d\phi \:d\theta \:$$
But this doesn't feel right at all. What's my problem?
The limits of integration are almost correct, but your integrand is wrong (and lacking parentheses). You should have $$\rho\cos\phi\cdot \rho^{-3} \cdot \underbrace{\rho^2\sin\phi\,d\rho\,d\phi\,d\theta}_{dV}.$$ Now, the lower limit for $\rho$ is wrong, as you should have the equation $\rho\cos\phi = 5/2$, so $\rho=\frac52\sec\phi$. (Think about why what you did makes no sense.)