Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y=25/x^2+25 and y=0 x=0 and x=5

To find the volume of the solid obtained by rotating the region enclosed by the curves \( y = \frac{25}{x^2 + 25} \), \( y = 0 \), \( x = 0 \), and \( x = 5 \) around the x-axis, we will use the **disk method**.

### Step-by-Step Solution

1. **Formula for Volume (Disk Method)**:
   The formula for the volume of a solid of revolution obtained by rotating a function \( y = f(x) \) around the x-axis between \( x = a \) and \( x = b \) is:
   \[
   V = \pi \int_{a}^{b} [f(x)]^2 \, dx
   \]
   In our case, \( f(x) = \frac{25}{x^2 + 25} \), \( a = 0 \), and \( b = 5 \).

2. **Square the function**:
   We need to square the function \( y = \frac{25}{x^2 + 25} \):
   \[
   \left( \frac{25}{x^2 + 25} \right)^2 = \frac{625}{(x^2 + 25)^2}
   \]

3. **Set up the integral**:
   The volume is given by:
   \[
   V = \pi \int_0^5 \frac{625}{(x^2 + 25)^2} \, dx
   \]

4. **Evaluate the integral**:
   We can solve this integral using substitution or recognize it as a standard form integral. Let's proceed with the integral:

   \[
   \int \frac{625}{(x^2 + 25)^2} \, dx
   \]

   This integral can be solved using a trigonometric substitution or known integral tables. The solution to this type of integral is:
   \[
   \int \frac{625}{(x^2 + 25)^2} \, dx = \frac{25x}{25^2 + x^2} + C
   \]

   Applying the limits from 0 to 5:

   \[
   V = \pi \left[ \frac{25x}{25^2 + x^2} \right]_0^5
   \]

   Substituting the values of \( x = 5 \) and \( x = 0 \) into the expression:
   \[
   V = \pi \left( \frac{25(5)}{25^2 + 5^2} - \frac{25(0)}{25^2 + 0^2} \right)
   \]
   Simplifying:
   \[
   V = \pi \left( \frac{125}{625 + 25} - 0 \right) = \pi \left( \frac{125}{650} \right) = \pi \left( \frac{25}{130} \right)
   \]
   \[
   V = \frac{25\pi}{130} = \frac{\pi}{5.2}
   \]

Thus, the volume of the solid is \( \frac{\pi}{5.2} \) cubic units.

Would you like further details on any step? Here are some related questions to consider:

1. How does the disk method differ from the washer method?
2. Can you solve the integral using trigonometric substitution?
3. How would the problem change if the region were rotated around the y-axis?
4. What are the other methods of calculating the volume of revolution?
5. How can the result be approximated numerically?

**Tip**: Always sketch the region and the solid to better visualize the volume of revolution.

Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y=25/(x^2+25), y=0, x=0, and x=5.

Explore how to calculate the volume of the solid obtained by rotating the curve y = 25/(x^2 + 25) about the x-axis, using the disk method. A detailed step-by-step solution is provided, with key concepts in calculus, integration, and solids of revolution. Suitable for advanced high school calculus students (Grades 11-12).

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