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 specified line.
y = x, y = 0, x = 4; about x = 9

We are tasked with setting up an integral to find the volume of the solid generated by rotating the region bounded by the curves \( y = x \), \( y = 0 \), and \( x = 4 \) about the line \( x = 9 \).

### Step 1: Identify the region
The region is bounded by:
- \( y = x \) (a straight line),
- \( y = 0 \) (the x-axis),
- \( x = 4 \) (a vertical line).

This describes a triangular region with vertices at points \( (0,0) \), \( (4,0) \), and \( (4,4) \).

### Step 2: Rotation about \( x = 9 \)
We are rotating the region about the vertical line \( x = 9 \). To apply the method of cylindrical shells, we use the shell method because we are rotating around a vertical line not on the region.

### Step 3: Shell Method Formula
The volume of a solid of revolution using the shell method is given by:

\[
V = 2\pi \int_a^b \text{(radius)} \times \text{(height)} \, dx
\]

- **Radius**: The distance from the curve to the axis of rotation. Here, the radius is \( 9 - x \), since we're rotating around \( x = 9 \).
- **Height**: The height of the shell is given by the function \( y = x \), so the height is \( x \).
- The bounds are from \( x = 0 \) to \( x = 4 \).

### Step 4: Set up the integral
Now we can set up the integral for the volume:

\[
V = 2\pi \int_0^4 (9 - x) \cdot x \, dx
\]

This integral represents the volume of the solid obtained by rotating the region about \( x = 9 \), but we

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 specified line. y = x, y = 0, x = 4; about x = 9.

Learn how to set up the integral for finding the volume of a solid of revolution obtained by rotating the region bounded by y = x, y = 0, and x = 4 about the line x = 9. This solution covers integral calculus concepts, including the cylindrical shell method and the setup of the appropriate integral for the volume.

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