Set up an integral that represents the volume generated by rotating the region bounded by the curves y = 4x - x^2 and the
X-axis about x = 7.

We are tasked with setting up an integral to find the volume generated by rotating the region bounded by the curve \( y = 4x - x^2 \) and the x-axis around the vertical line \( x = 7 \).

### Step 1: Analyze the region

The region is bounded by:
- The curve \( y = 4x - x^2 \)
- The x-axis, where \( y = 0 \)

To find the points of intersection between the curve and the x-axis, set \( y = 4x - x^2 = 0 \):
\[
x(4 - x) = 0 \implies x = 0 \quad \text{or} \quad x = 4
\]
So, the region is between \( x = 0 \) and \( x = 4 \).

### Step 2: Use the method of cylindrical shells

Since we are rotating around \( x = 7 \), a vertical axis, we will use the **method of cylindrical shells**. The formula for the volume using cylindrical shells is:
\[
V = 2\pi \int_a^b ( \text{radius} )( \text{height} ) \, dx
\]
- **Radius**: The distance from a point on the curve at \( x \) to the axis of rotation at \( x = 7 \), which is \( 7 - x \).
- **Height**: The height of the shell is the value of \( y \), which is \( 4x - x^2 \).

### Step 3: Set up the integral

Now, we integrate from \( x = 0 \) to \( x = 4 \):
\[
V = 2\pi \int_0^4 (7 - x)(4x - x^2) \, dx
\]

This is the integral that represents the volume of the solid generated by rotating the region about \( x = 7 \).

Would you like to proceed with evaluating the integral or need further details?

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**Relative Questions:**
1. How do you solve the integral using expansion and substitution methods?
2. What is the difference between cylindrical shells and the disk method?
3. How would the setup change if we rotated around a different vertical line, say \( x = -1 \)?
4. Can this volume be calculated using the disk/washer method instead of cylindrical shells?
5. How does the integral setup change if the region is rotated about the y-axis?

**Tip:** For problems involving rotation around a vertical or horizontal axis, always visualize the shape first to determine the most suitable method (cylindrical shells or disk/washer).

Set up an integral that represents the volume generated by rotating the region bounded by the curves y = 4x - x^2 and the X-axis about x = 7.

This guide explains how to set up the integral to find the volume of a solid generated by rotating the region bounded by the curve y = 4x - x^2 and the x-axis around the vertical line x = 7 using the cylindrical shells method. The problem explores advanced calculus concepts and is suitable for Grades 11-12 or early college students.

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