The OCR output from the image is not very clear. It seems like there were errors in recognizing the text. Let's manually interpret the visible mathematical content in the image:

The problem statement appears to be:

"Evaluate the triple integral $\iiint_E f(x, y, z) \, dV$, where $f(x, y, z) = x^2 + y^2 + z^2$, over the part of the ball $x^2 + y^2 + z^2 \leq 9$ defined by $z \geq 2$."

To solve this triple integral:

1. Transform to Spherical Coordinates: $x = \rho \sin\phi \cos\theta, \quad y = \rho \sin\phi \sin\theta, \quad z = \rho \cos\phi$ with the Jacobian determinant $\rho^2 \sin\phi$.

2. Limits of Integration:

   • $\rho$ ranges from 0 to 3 (since the radius of the ball is 3).
   • $\phi$ ranges from 0 to $\cos^{-1}\left(\frac{2}{3}\right)$ (the angle corresponding to $z \geq 2$).
   • $\theta$ ranges from 0 to $2\pi$.

3. Integral Setup: $\iiint_E f(x, y, z) \, dV = \int_{0}^{2\pi} \int_{0}^{\cos^{-1}(2/3)} \int_{0}^{3} \left( \rho^2 \sin^2\phi \cos^2\theta + \rho^2 \sin^2\phi \sin^2\theta + \rho^2 \cos^2\phi \right) \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$ Simplifying the integrand: $\int_{0}^{2\pi} \int_{0}^{\cos^{-1}(2/3)} \int_{0}^{3} \rho^4 \left( \sin^2\phi (\cos^2\theta + \sin^2\theta) + \cos^2\phi \right) \sin\phi \, d\rho \, d\phi \, d\theta$ Since $\cos^2\theta + \sin^2\theta = 1$: $\int_{0}^{2\pi} \int_{0}^{\cos^{-1}(2/3)} \int_{0}^{3} \rho^4 \left( \sin^2\phi + \cos^2\phi \right) \sin\phi \, d\rho \, d\phi \, d\theta$ $= \int_{0}^{2\pi} \int_{0}^{\cos^{-1}(2/3)} \int_{0}^{3} \rho^4 \sin\phi \, d\rho \, d\phi \, d\theta$

4. Separating Integrals:

   • The integral with respect to $\rho$: $\int_{0}^{3} \rho^4 \, d\rho = \left. \frac{\rho^5}{5} \right|_0^3 = \frac{3^5}{5} = \frac{243}{5}$
   • The integral with respect to $\phi$: $\int_{0}^{\cos^{-1}(2/3)} \sin\phi \, d\phi = \left. -\cos\phi \right|_0^{\cos^{-1}(2/3)} = -\cos\left(\cos^{-1}\left(\frac{2}{3}\right)\right) + \cos(0) = -\frac{2}{3} + 1 = \frac{1}{3}$
   • The integral with respect to $\theta$: $\int_{0}^{2\pi} d\theta = 2\pi$

5. Final Calculation: $\iiint_E f(x, y, z) \, dV = \left( \frac{243}{5} \right) \left( \frac{1}{3} \right) \left( 2\pi \right) = \frac{243 \times 2\pi}{15} = \frac{486\pi}{15} = \frac{162\pi}{5}$

Therefore, the value of the triple integral is $\boxed{\frac{162\pi}{5}}$.

Would you like any more details or have any questions?

Here are 8 related questions for further practice:

1. Evaluate $\iiint_E 1 \, dV$ over the same region.
2. Evaluate the integral $\iiint_E x \, dV$ over the same region.
3. What is the volume of the part of the ball $x^2 + y^2 + z^2 \leq 9$ defined by $z \geq 2$?
4. Compute $\iiint_E e^{x^2 + y^2 + z^2} \, dV$ over the same region.
5. Evaluate $\iiint_E z \, dV$ over the same region.
6. Evaluate the integral $\iiint_E \sin(x^2 + y^2 + z^2) \, dV$ over the same region.
7. How would the integral change if the region is defined by $x^2 + y^2 + z^2 \leq 16$ and $z \geq 2$?
8. Find the centroid of the region defined by $x^2 + y^2 + z^2 \leq 9$ and $z \geq 2$.

Tip: Always double-check the limits of integration, especially when switching coordinate systems.