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The problem asks to use the method of cylindrical shells to find the volume of the solid generated by rotating the region bounded by \( y = x^3 \), \( y = 8 \), and \( x = 0 \) about \( x = 5 \).

To solve this using the method of cylindrical shells, follow these steps:

### Step 1: Formula for Volume Using Cylindrical Shells
The volume \( V \) of a solid generated by rotating a region about a vertical axis (in this case, \( x = 5 \)) is given by:
\[
V = 2\pi \int_{a}^{b} ( \text{radius} )( \text{height} ) \, dx
\]
where:
- The **radius** is the distance from the axis of rotation, \( x = 5 \), which is \( r(x) = 5 - x \).
- The **height** is the function describing the region's height, which in this case is \( h(x) = y = x^3 \).

### Step 2: Identify the Bounds of Integration
From the equations given:
- \( y = x^3 \) reaches \( y = 8 \) at \( x = 2 \), since \( (2)^3 = 8 \).
- Therefore, the bounds of integration will be from \( x = 0 \) to \( x = 2 \).

### Step 3: Set Up the Integral
Now, set up the integral for the volume:
\[
V = 2\pi \int_{0}^{2} (5 - x)(x^3) \, dx
\]
where:
- \( (5 - x) \) is the radius,
- \( x^3 \) is the height.

### Step 4: Solve the Integral (optional if the problem just asks to set it up)
Expanding the integrand gives:
\[
V = 2\pi \int_{0}^{2} (5x^3 - x^4) \, dx
\]
The rest involves computing this integral step by step if needed.

Would you like further explanation on solving this integral or any related step?

### Follow-up Questions:
1. What is the significance of using cylindrical shells versus the disk/washer method?
2. How do you determine the limits of integration in such problems?
3. Why is the distance from the axis of rotation \( r(x) = 5 - x \)?
4. How would the setup differ if we rotated about a different axis, such as \( x = 0 \)?
5. Can this method be applied if the functions involved were more complex?

**Tip:** Always double-check the limits of integration to ensure they match the intersection points of the functions.

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by y = x^3, y = 8, and x = 0 about x = 5. Set up an integral that can be used to determine the volume V of the solid.

Learn how to set up an integral to find the volume generated by rotating the region bounded by y = x^3, y = 8, and x = 0 about the vertical axis x = 5 using the method of cylindrical shells. This problem covers advanced calculus concepts related to volumes of solids of revolution.

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