Here is the question:
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified x-axis.
Here are the parameters:
$$y = 4x - x^2, y = 3\text {; about }x = 1$$
So here are my choices for radius and height:
radius = $x - 1$
height = $4x - x^2 - 3$
So...is this setup right?
$$ \int_1^3 2 \pi (x - 1) ( 4x - x^2 - 3)\, dx$$
You are correct.
The volume of revolution is presented by $$\int_1^3 2 \pi (x - 1) ( 4x - x^2 - 3)\, dx$$
and you have explained it well in detail.