Find the volume *V* of the solid obtained by rotating the region bounded by the given curves about the specified line.

*y* = 5(25 -x^2)^1/2

,   *y* = 0,   *x* = 2,   *x* = 4;

about the *x*-axis

The problem asks to find the volume of the solid obtained by rotating the region bounded by the curves \( y = 5 \sqrt{25 - x^2} \), \( y = 0 \), \( x = 2 \), and \( x = 4 \) around the x-axis. This can be solved using the method of disks.

### Step-by-Step Solution:

1. **Set up the disk method formula:**

   The volume \( V \) of a solid of revolution generated by rotating a curve around the x-axis is given by the formula:

   \[
   V = \pi \int_{a}^{b} [f(x)]^2 \, dx
   \]
   
   where \( f(x) \) is the function being rotated, and \( [a, b] \) are the limits of integration along the x-axis.

2. **Identify the function and limits:**
   
   - The curve is \( y = 5 \sqrt{25 - x^2} \), so \( f(x) = 5 \sqrt{25 - x^2} \).
   - The region is bounded by \( x = 2 \) and \( x = 4 \), so the limits of integration are \( a = 2 \) and \( b = 4 \).

3. **Set up the integral:**

   The volume integral becomes:

   \[
   V = \pi \int_{2}^{4} [5 \sqrt{25 - x^2}]^2 \, dx
   \]
   
   Simplifying the square of the function:

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

4. **Solve the integral:**

   Now distribute the 25:

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

   Break the integral into two parts:

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

   First, solve \( \int_{2}^{4} 25 \, dx \):

   \[
   \int_{2}^{4} 25 \, dx = 25(x) \bigg|_2^4 = 25(4 - 2) = 50
   \]

   Now, solve \( \int_{2}^{4} x^2 \, dx \):

   \[
   \int_{2}^{4} x^2 \, dx = \frac{x^3}{3} \bigg|_2^4 = \frac{4^3}{3} - \frac{2^3}{3} = \frac{64}{3} - \frac{8}{3} = \frac{56}{3}
   \]

5. **Substitute back into the volume formula:**

   \[
   V = 25 \pi \left[ 50 - \frac{56}{3} \right]
   \]

   Simplify:

   \[
   V = 25 \pi \left[ \frac{150}{3} - \frac{56}{3} \right] = 25 \pi \cdot \frac{94}{3}
   \]

   Finally:

   \[
   V = \frac{2350 \pi}{3}
   \]

### Final Answer:

The volume of the solid is:

\[
V = \frac{2350 \pi}{3} \, \text{cubic units}
\]

Would you like further details on any of the steps?

---

**Related Questions:**

1. What is the volume of a solid generated by rotating the curve \( y = \sqrt{25 - x^2} \) around the x-axis for different bounds?
2. How do you apply the shell method for the same problem?
3. What happens if we rotate the region around the y-axis instead of the x-axis?
4. How does the volume change if the region is rotated around a line other than the x-axis?
5. What are other methods to find the volume of solids of revolution?

**Tip:** Always check if the integral can be simplified before applying advanced techniques.

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

 y = 5(25 - x^2)^1/2 ,   y = 0,   x = 2,   x = 4; about the x-axis.

Find the volume of the solid generated by rotating the curve y = 5√(25 - x²) around the x-axis, using the disk method. Step-by-step solution provided, covering concepts from calculus and solids of revolution, with integral calculation. Suitable for advanced high school and university students.

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