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To find the volume of the solid of revolution created by rotating the region bounded by \( f(x) = -8 \sin(x) \), \( x = \pi \), \( x = 2\pi \), and \( y = 0 \) around the \( y \)-axis, we’ll use the method of cylindrical shells.

### Solution Steps

1. **Identify the Shell Radius and Height**:  
   - Since the rotation is about the \( y \)-axis, the radius of a shell at \( x \) will be \( x \).
   - The height of the shell at \( x \) is given by \( |f(x)| = 8 \sin(x) \).

2. **Set Up the Integral for Volume**:  
   The volume \( V \) of a solid of revolution using cylindrical shells is:
   \[
   V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx
   \]
   In this case:
   \[
   V = \int_{\pi}^{2\pi} 2\pi \cdot x \cdot 8 \sin(x) \, dx
   \]
   Simplify:
   \[
   V = \int_{\pi}^{2\pi} 16\pi x \sin(x) \, dx
   \]

3. **Integrate \( 16\pi x \sin(x) \)** Using Integration by Parts:
   To solve \( \int 16\pi x \sin(x) \, dx \), let:
   - \( u = x \) and \( dv = \sin(x) \, dx \).
   - Then \( du = dx \) and \( v = -\cos(x) \).
   
   Applying integration by parts:
   \[
   \int x \sin(x) \, dx = -x \cos(x) + \int \cos(x) \, dx = -x \cos(x) + \sin(x)
   \]

   Thus,
   \[
   V = 16\pi \left( \left[ -x \cos(x) + \sin(x) \right]_{\pi}^{2\pi} \right)
   \]

4. **Evaluate the Integral**:
   Substitute \( x = 2\pi \) and \( x = \pi \) into the expression, and simplify to get the exact volume.

Let me know if you would like further details on each step!

### Follow-Up Questions

1. How does rotating around the \( y \)-axis differ from rotating around the \( x \)-axis?
2. What other methods can be used to find the volume of a solid of revolution?
3. Can the washer method also be applied here? Why or why not?
4. How would the integral change if the function was rotated around a different line?
5. What happens if the bounds are extended beyond \( [\pi, 2\pi] \)?

### Tip
In problems involving volumes of revolution, carefully choose the axis of rotation and set up integrals with the right method to simplify computation.

The region bounded by f(x) = -8sin(x), x = pi, x = 2pi, and y = 0 is rotated about the y-axis. Find the volume of the solid of revolution.

Learn how to find the volume of a solid of revolution formed by rotating f(x) = -8sin(x) around the y-axis, with a detailed step-by-step solution using the cylindrical shells method. Suitable for advanced calculus students.

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