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To set up an integral using the **shell method** for the volume of the solid formed by rotating the region around the **x-axis**, we follow these steps:

### Given curves and information:
- \( y = x^3 \)
- \( y = 64 \)
- \( x = 0 \)

### Shell Method Setup:
The shell method involves using vertical slices of the region and rotating them around the x-axis. The formula for the volume using the shell method is:

\[
V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dy
\]

Where:
- **radius** is the distance from the x-axis, which is \( y \),
- **height** is the horizontal distance between the curves, which is \( x = \sqrt[3]{y} \).

### Limits of Integration:
- The lower bound is \( y = 0 \) (where \( x = 0 \)),
- The upper bound is \( y = 64 \).

### Integral Setup:
\[
V = 2\pi \int_{0}^{64} y \times \left(\sqrt[3]{y}\right) \, dy
\]

This is the setup for the volume of the solid using the shell method, but we are not evaluating the integral as per the problem's instructions.

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Would you like to go over any step in more detail, or need further clarification?

Here are five related questions to extend the topic:
1. How would the method change if the region were rotated about the y-axis instead?
2. What would the limits of integration be if the region was bounded by a different upper function?
3. Can you apply the disk method to this problem and compare the setup with the shell method?
4. How would the volume change if the region was rotated around a different line, such as \( y = 10 \)?
5. What happens if we rotate the region about the line \( x = -1 \)? How would the shell method change?

**Tip:** When using the shell method, the height function is typically a horizontal distance (a function of \( y \)) if rotating around the x-axis.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis using the shell method.

Set up the integral for finding the volume of a solid formed by rotating the region bounded by y = x³, y = 64, and x = 0 around the x-axis. The problem involves the shell method, with integration limits from y = 0 to y = 64. Suitable for calculus students in Grades 11-12 or college-level courses.

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