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The given problem is a triple integral over the region \( \Omega \), where \( \Omega \) is defined by \( 4 \leq x^2 + y^2 + z^2 \leq 16 \), and the integral to evaluate is:

\[
\iiint_\Omega \frac{1}{\sqrt{x^2 + y^2 + z^2}} \, dx \, dy \, dz.
\]

### Approach to Solve
The region \( \Omega \) is a spherical shell bounded by two spheres with radii \( 2 \) and \( 4 \) (since \( 4 = 2^2 \) and \( 16 = 4^2 \)). We switch to spherical coordinates, where:
- \( x = \rho \sin\phi \cos\theta \),
- \( y = \rho \sin\phi \sin\theta \),
- \( z = \rho \cos\phi \),
- The volume element \( dx \, dy \, dz = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \).

The given function \( \frac{1}{\sqrt{x^2 + y^2 + z^2}} \) becomes \( \frac{1}{\rho} \) in spherical coordinates. The integral now becomes:

\[
\iiint_\Omega \frac{1}{\sqrt{x^2 + y^2 + z^2}} \, dx \, dy \, dz = \int_0^{2\pi} \int_0^\pi \int_2^4 \frac{1}{\rho} \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta.
\]

Simplify the integrand:
\[
\frac{1}{\rho} \rho^2 = \rho.
\]

Thus, the integral becomes:
\[
\int_0^{2\pi} \int_0^\pi \int_2^4 \rho \sin\phi \, d\rho \, d\phi \, d\theta.
\]

### Step 1: Evaluate the \( \rho \)-Integral
\[
\int_2^4 \rho \, d\rho = \left[ \frac{\rho^2}{2} \right]_2^4 = \frac{4^2}{2} - \frac{2^2}{2} = \frac{16}{2} - \frac{4}{2} = 8 - 2 = 6.
\]

### Step 2: Evaluate the \( \phi \)-Integral
\[
\int_0^\pi \sin\phi \, d\phi = \left[ -\cos\phi \right]_0^\pi = -\cos\pi + \cos0 = -(-1) + 1 = 2.
\]

### Step 3: Evaluate the \( \theta \)-Integral
\[
\int_0^{2\pi} 1 \, d\theta = \left[ \theta \right]_0^{2\pi} = 2\pi.
\]

### Combine the Results
Multiply the results from each step:
\[
6 \cdot 2 \cdot 2\pi = 24\pi.
\]

### Final Answer:
\[
\boxed{24\pi}
\]

Let me know if you need further clarifications or have questions!

Calculate the triple integral \( \iiint_\Omega \frac{1}{\sqrt{x^2 + y^2 + z^2}} \, dx \, dy \, dz \), where \( \Omega \) is defined as \( 4 \leq x^2 + y^2 + z^2 \leq 16 \).

Learn how to evaluate the triple integral \( \iiint_\Omega \frac{1}{\sqrt{x^2 + y^2 + z^2}} \, dx \, dy \, dz \) over a spherical shell region \( 4 \leq x^2 + y^2 + z^2 \leq 16 \). This solution uses spherical coordinates to simplify the integral, with detailed steps for each part of the evaluation.

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