Area and Volume Relationship

Two parallel lines $P$ and $Q$ are drawn arbitrarily on the plane. For each line, two points are positioned $D$ units apart, where $D$ is the shortest distance between $P$ and $Q$ . Then, the points are connected to form the non-self-intersecting quadrilateral (as in the example below). If we revolve the quadrilateral about the line of revolution (also parallel to both $P$ and $Q$ ), then we obtain either a cylinder or a hollow cylinder of volume $V_{\text{cylinder}}$ .

The revolution setup of non-self-intersecting figure.

Now, consider forming the self-intersecting figure, as shown below. If we follow the same steps for this, what is the volume of the second solid in terms of $V_{\text{cylinder}}?$

Details and Assumptions:

• The line of revolution doesn't lie between lines $P$ and $Q$ . However, it can be collinear with either of these two lines.
• The line and point configurations are the same for both cases.

The revolution setup of self-intersecting figure.