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We are asked to find the volume of the solid obtained by rotating the region enclosed by the curves:

\[
y = \frac{16}{x^2 + 16}, \quad y = 0, \quad x = 0, \quad \text{and} \quad x = 4
\]

about the x-axis.

### Step-by-step Solution:

To solve this, we can use the **disk method** since the region is being rotated about the x-axis. The volume \( V \) of the solid of revolution is given by:

\[
V = \pi \int_{a}^{b} \left( f(x) \right)^2 \, dx
\]

Here, \( f(x) = \frac{16}{x^2 + 16} \), and the limits of integration are from \( x = 0 \) to \( x = 4 \).

Thus, the volume \( V \) is:

\[
V = \pi \int_{0}^{4} \left( \frac{16}{x^2 + 16} \right)^2 \, dx
\]

### Simplifying:

First, we square the function:

\[
\left( \frac{16}{x^2 + 16} \right)^2 = \frac{256}{(x^2 + 16)^2}
\]

So the integral becomes:

\[
V = \pi \int_{0}^{4} \frac{256}{(x^2 + 16)^2} \, dx
\]

This is a standard integral, and we can solve it using substitution and known formulas from integral tables.

### Solution of the Integral:

To proceed with the calculation, I will now solve the integral using substitution. Let me compute the result for you.The volume of the solid obtained by rotating the region about the x-axis is:

\[
V = \frac{\pi(2 + \pi)}{2}
\]

Would you like further details or a breakdown of any specific step?

Here are 5 related questions for deeper understanding:
1. How would the volume change if the region were rotated about the y-axis instead?
2. What would the volume be if the boundaries of \( x \) were from 0 to a different value (e.g., \( x = 6 \))?
3. Can you calculate the area under the curve \( y = \frac{16}{x^2 + 16} \) from \( x = 0 \) to \( x = 4 \)?
4. What if the function \( y = \frac{16}{x^2 + 16} \) had a different constant in the numerator?
5. How does the disk method compare to the washer method in terms of application?

**Tip**: Always check the boundaries of integration carefully, especially when solving problems involving rotation or revolved solids.

Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = 16 / (x^2 + 16), y = 0, x = 0, and x = 4.

This problem involves calculating the volume of a solid obtained by rotating a region enclosed by curves around the x-axis using the disk method. The integral is solved step-by-step with detailed explanations, ideal for college-level calculus students.

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