Find the volume of the solid generated by revolving the region enclosed by the triangle with vertices left parenthesis 4 comma 3 right parenthesis(4,3), left parenthesis 4 comma 4 right parenthesis(4,4), and left parenthesis 7 comma 4 right parenthesis(7,4) about the y-axis.

So we have three points A(4,3), B(4,4) and C(7,4). 
We introduce a fourth point D(7,3).
We obtain a rectangle ABCD. By rotating the rectangle, 
around the y axis, we get two cylinders. Denote the 
first cylinder C1 and the inner cylinder with vertices AB by C2. 
Computing the volume of C1:

Computing the volume of the cylinder C2:
 
In order to find the volume of the hollow tube, simply compute the difference of the two obtained values like this:

Now, in order to find the volume of the solid generated by revolving the region enclosed by the triangle, simply divide the previous value over two like this:


Answer 

From the sketch of the required triangle, the volume of the solid generated by revolving the region enclosed by the triangle with vertices (4, 3), (4, 4), and (7, 4) about the y-axis is given by