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