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Here are the steps to solve the problem:

We are asked to find the volume of the solid obtained by rotating the region bounded by \( y = \sqrt{25 - x^2} \), \( y = 0 \), \( x = 2 \), and \( x = 4 \) about the x-axis. This is a problem involving the method of disks or washers.

### Step 1: Formula for Volume of Revolution

The volume of a solid of revolution about the x-axis can be calculated using the formula:

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

where \( f(x) \) is the function being revolved, and \( [a, b] \) are the limits of integration.

### Step 2: Apply the Function and Limits

The function is \( f(x) = \sqrt{25 - x^2} \), and the limits of integration are from \( x = 2 \) to \( x = 4 \).

\[
V = \pi \int_{2}^{4} (\sqrt{25 - x^2})^2 \, dx
\]

Simplify the integrand:

\[
V = \pi \int_{2}^{4} (25 - x^2) \, dx
\]

### Step 3: Integrate the Function

Now, integrate the function:

\[
V = \pi \left[ 25x - \frac{x^3}{3} \right]_{2}^{4}
\]

### Step 4: Evaluate the Definite Integral

Now, plug in the limits of integration:

\[
V = \pi \left( \left[ 25(4) - \frac{4^3}{3} \right] - \left[ 25(2) - \frac{2^3}{3} \right] \right)
\]

First, calculate each term:

For \( x = 4 \):

\[
25(4) = 100, \quad \frac{4^3}{3} = \frac{64}{3}
\]

For \( x = 2 \):

\[
25(2) = 50, \quad \frac{2^3}{3} = \frac{8}{3}
\]

Now, substitute these into the equation:

\[
V = \pi \left( \left[ 100 - \frac{64}{3} \right] - \left[ 50 - \frac{8}{3} \right] \right)
\]

Simplify:

\[
V = \pi \left( \frac{300}{3} - \frac{64}{3} - \left( \frac{150}{3} - \frac{8}{3} \right) \right)
\]

\[
V = \pi \left( \frac{236}{3} - \frac{142}{3} \right)
\]

\[
V = \pi \cdot \frac{94}{3}
\]

### Step 5: Final Answer

Thus, the volume is:

\[
V = \frac{94\pi}{3} \ \text{cubic units}
\]

Find the volume of the solid obtained by rotating the region bounded by y = sqrt(25 - x^2), y = 0, x = 2, and x = 4 about the x-axis.

Learn how to calculate the volume of a solid obtained by rotating the region between y = sqrt(25 - x^2), y = 0, x = 2, and x = 4 about the x-axis. This step-by-step guide covers the method of washers and involves integration techniques suitable for Grade 12 or College-level Calculus students.

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