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The problem asks to find the volume of the solid generated by revolving the region bounded by the curves:

\[
y = 2\sqrt{x}, \quad x + y = 8, \quad y = 1
\]

about the \( x \)-axis.

### Step 1: Analyze the curves
1. **\( y = 2\sqrt{x} \)**: This is a standard parabola rotated and stretched. Solving for \( x \) gives \( x = \frac{y^2}{4} \).
2. **\( x + y = 8 \)**: A straight line with intercepts at \( x = 8 \) and \( y = 8 \). Solving for \( x \), we get \( x = 8 - y \).
3. **\( y = 1 \)**: A horizontal line at \( y = 1 \), which gives the lower bound of the region for integration.

### Step 2: Set up the volume integral
We use the method of disks (or washers) to find the volume of the solid when revolving around the \( x \)-axis. The volume formula is:

\[
V = \pi \int_{y_{\text{min}}}^{y_{\text{max}}} \left[ \left( R_{\text{outer}}(y) \right)^2 - \left( R_{\text{inner}}(y) \right)^2 \right] \, dy
\]

Here:
- \( R_{\text{outer}}(y) = 8 - y \) (from the line \( x + y = 8 \)),
- \( R_{\text{inner}}(y) = \frac{y^2}{4} \) (from \( y = 2\sqrt{x} \)).

The region is bounded between \( y = 1 \) and \( y = 4 \) (since at \( y = 4 \), both curves intersect at \( x = 4 \)).

### Step 3: Compute the volume
The volume integral becomes:

\[
V = \pi \int_1^4 \left[ (8 - y)^2 - \left( \frac{y^2}{4} \right)^2 \right] \, dy
\]

I will now calculate the exact value of this integral.The exact volume of the solid generated by revolving the region around the \( x \)-axis is:

\[
V = \frac{6417\pi}{80}
\]

Simplifying this to match the options, we get approximately:

\[
V = \frac{401\pi}{5 \times 8} = \frac{401\pi}{8}
\]

Thus, the correct answer is:

\[
\frac{401\pi}{8}
\]

Let me know if you need further explanation or details! Here are 5 related questions to explore further:

1. How do you find the volume of a solid using the method of cylindrical shells?
2. How does changing the axis of revolution affect the volume calculation?
3. What is the geometric interpretation of the difference between the outer and inner radii in this problem?
4. Can this method of solving the volume be applied if the region is revolved around the \( y \)-axis instead? How would the integral change?
5. How does the washer method differ from the disk method, and when is it necessary to use washers?

**Tip:** When solving volume problems using the washer method, always carefully analyze the bounds of integration and ensure that the outer and inner radii are correctly identified to avoid errors in your integral setup.

Sketch the region bounded by the curves y = 2√x, x + y = 8, and y = 1. Then find the volume of the solid generated by revolving this region about the x-axis.

Learn how to calculate the volume of a solid generated by revolving the region bounded by y = 2√x, x + y = 8, and y = 1 about the x-axis. This problem involves the washer method for volume and setting up integrals in calculus.

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