Use the disk method *or* the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line.

y= 10/(x^2), y = 0, x= 1, x = 5. Revolve it around the line y=10

To find the volume of the solid generated by revolving the region bounded by the curves \( y = \frac{10}{x^2} \), \( y = 0 \), \( x = 1 \), and \( x = 5 \) about the line \( y = 10 \), we can use either the disk method or the shell method.

### Approach: Shell Method
Since the axis of rotation, \( y = 10 \), is horizontal, and the region is described in terms of \( y = f(x) \), the shell method is more convenient here. This involves integrating cylindrical shells formed by revolving the region around the line \( y = 10 \).

The formula for the volume using the shell method is:
\[
V = 2\pi \int_{x=a}^{x=b} ( \text{radius} ) (\text{height}) \, dx
\]
- **Radius**: The distance from a point on the curve to the axis of rotation \( y = 10 \). For the curve \( y = \frac{10}{x^2} \), this distance is \( 10 - \frac{10}{x^2} \).
- **Height**: The height of the shell is given by the function \( y = \frac{10}{x^2} \).

Thus, the volume is given by:
\[
V = 2\pi \int_{1}^{5} \left( 10 - \frac{10}{x^2} \right) \cdot \frac{10}{x^2} \, dx
\]

Now, let's compute this integral to find the volume.The volume of the solid generated by revolving the region bounded by the curves \( y = \frac{10}{x^2} \), \( y = 0 \), \( x = 1 \), and \( x = 5 \) around the line \( y = 10 \) is:

\[
V = \frac{1408\pi}{15} \text{ cubic units.}
\]

Would you like more details or further clarification on the method used?

Here are some related questions you might find useful:
1. How do you set up the disk method for this problem?
2. Can this problem be solved using polar coordinates?
3. What is the geometric interpretation of cylindrical shells?
4. How does the choice of axis of rotation affect the method?
5. How would the volume change if the region were revolved around the x-axis?

**Tip:** When using the shell method, always visualize the height and radius of the cylindrical shells to correctly set up the integral.

Learn how to find the volume of the solid generated by revolving the region bounded by the curves y = 10/(x^2), y = 0, x = 1, and x = 5 around the line y = 10 using the shell method in calculus. Detailed step-by-step solution provided.

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