Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = (25 − x^2)^1/5
,   y = 0,   x = 2,   x = 4;
about the x-axis

We are tasked with finding the volume \( V \) of the solid obtained by rotating the region bounded by the curves about the x-axis using the **disk method**. The general formula for the volume of a solid of revolution using the disk method is:

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

### Step-by-step solution:

1. **Identify the boundaries:**
   - The function: \( y = (25 - x^2)^{1/5} \)
   - The region is bounded by \( y = 0 \), \( x = 2 \), and \( x = 4 \).

2. **Set up the volume integral:**
   We will rotate the region about the x-axis, so the radius of the disk at each point \( x \) is given by \( y = (25 - x^2)^{1/5} \). The volume is:

   \[
   V = \pi \int_{2}^{4} [(25 - x^2)^{1/5}]^2 \, dx
   \]

3. **Simplify the integrand:**
   The integrand simplifies as follows:

   \[
   (25 - x^2)^{2/5}
   \]

   So the volume integral becomes:

   \[
   V = \pi \int_{2}^{4} (25 - x^2)^{2/5} \, dx
   \]

4. **Evaluate the integral:**
   To solve this integral, we can apply substitution and numerical methods if needed. I'll compute the integral now.

The volume \( V \) of the solid obtained by rotating the region bounded by the given curves about the x-axis is approximately:

\[
V \approx 18.77 \, \text{cubic units}
\]

Would you like more details about the steps or the methods used?

Here are some related questions you might want to explore:
1. How would the volume change if the region was rotated about the y-axis?
2. What would be the volume if the bounds were extended to \( x = 1 \) and \( x = 4 \)?
3. Can the shell method be applied to this problem, and how would it differ from the disk method?
4. How does the exponent \( \frac{2}{5} \) in the integrand affect the shape of the solid?
5. What are other methods to approximate the volume of a solid of revolution?

**Tip:** When dealing with complex integrals, sometimes numerical methods (like the one used here) are necessary to compute results that aren't easily solvable by hand.

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = (25 − x^2)^1/5, y = 0, x = 2, x = 4; about the x-axis.

This solution explains how to find the volume of the solid obtained by rotating the region bounded by y = (25 − x^2)^{1/5}, y = 0, x = 2, and x = 4 about the x-axis using the disk method. Learn about setting up and solving integrals for volumes of revolution.

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