Evaluate the following integral in spherical coordinates. $\iiint_{D}\left(x^{2}+y^{2}+z^{2}\right)^{5 / 2} d V ; D$ is the unit ball centered at the origin Set up the triple integral using spherical coordinates that should be used to evaluate the given integral as efficiently as possible. Use increasing limits of integration. (Type an exact answer, using $\pi$ as needed.)

Step 1 ：First, we need to convert the given integral into spherical coordinates. In spherical coordinates, \(x = r \sin(\theta) \cos(\phi)\), \(y = r \sin(\theta) \sin(\phi)\), and \(z = r \cos(\theta)\). Therefore, \(x^2 + y^2 + z^2 = r^2\).

Step 2 ：The volume element in spherical coordinates is \(dV = r^2 \sin(\theta) dr d\theta d\phi\).

Step 3 ：The region D is the unit ball centered at the origin. In spherical coordinates, this corresponds to \(0 \leq r \leq 1\), \(0 \leq \theta \leq \pi\), and \(0 \leq \phi \leq 2\pi\).

Step 4 ：Substituting these into the integral, we get \(\iiint_{D} r^{10} \sin(\theta) dr d\theta d\phi\).

Step 5 ：We can now evaluate this integral by splitting it into three separate integrals: \(\int_{0}^{1} r^{10} dr \int_{0}^{\pi} \sin(\theta) d\theta \int_{0}^{2\pi} d\phi\).

Step 6 ：The first integral evaluates to \(\frac{1}{11}\). The second integral evaluates to \(2\). The third integral evaluates to \(2\pi\).

Step 7 ：Multiplying these together, we get the final result: \(\frac{4\pi}{11}\).

Step 8 ：Therefore, the value of the given integral over the unit ball centered at the origin is \(\boxed{\frac{4\pi}{11}}\).