Set up the integral that would give the volume *V* generated by rotating the region bounded by the given curves about the *y*-axis.

*y* = *x*3,    *y* = 0,    *x* = 4

To set up the integral that gives the volume \( V \) generated by rotating the region bounded by the curves \( y = x^3 \), \( y = 0 \), and \( x = 4 \) about the \( y \)-axis, we can use the method of cylindrical shells.

### Step 1: Identify the region to be rotated
The region is bounded by:
- \( y = x^3 \) (the curve)
- \( y = 0 \) (the x-axis)
- \( x = 4 \) (vertical line)

### Step 2: Use the method of cylindrical shells
For rotation about the \( y \)-axis, the volume \( V \) can be calculated using the formula:
\[
V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\]
where \( f(x) \) is the function being rotated, \( x \) is the radius of the shell, and \( [a, b] \) are the limits of integration in terms of \( x \).

### Step 3: Set up the integral
Given:
- \( f(x) = x^3 \)
- The limits of integration are from \( x = 0 \) to \( x = 4 \)

The integral becomes:
\[
V = 2\pi \int_{0}^{4} x \cdot x^3 \, dx
\]
\[
V = 2\pi \int_{0}^{4} x^4 \, dx
\]

This integral represents the volume of the solid generated by rotating the region about the \( y \)-axis.

Would you like me to calculate the integral or provide further details?

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Here are some related questions to consider:
1. How would the setup change if the region was rotated about the \( x \)-axis instead of the \( y \)-axis?
2. What if the rotation was about the line \( x = 4 \) instead of the \( y \)-axis?
3. How does the method of disks or washers differ from the method of cylindrical shells?
4. What is the geometric interpretation of the integral used in the shell method?
5. How would the integral change if the curve was \( y = x^2 \) instead of \( y = x^3 \)?

**Tip:** When setting up an integral for volume, always consider the axis of rotation carefully, as it determines whether to use the disk/washer method or the shell method.

Set up the integral that would give the volume V generated by rotating the region bounded by the given curves about the y-axis. y = x^3, y = 0, x = 4

Learn how to set up the integral for calculating the volume generated by rotating the region bounded by the curves y = x^3, y = 0, and x = 4 about the y-axis. This example uses the cylindrical shells method and provides a step-by-step solution suitable for students in Grades 11-12.

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