I know the basic idea of disc / washer / cylindrical shell methods, to find the volume of solids generated : be it a single function, or the bounded region between two functions.
The trouble I am not able to resolve is, when the axis of rotation itself passes through the region bounded between the curves. The particular issue I have here is, how to identify that which one out of the two functions shall define the volume, since both are mutually going to revolve about the same line.
For ex, consider the area bounded between the curves $y = x^2 - 3$ and $y = 2x$, which extends in the interval $-1<x<3$. Now let this region be revolved about the line $y=1$. To the naked eye, it is a bit difficult to decipher that which function has the higher ordinate value, and hence should be considered for the volume generated. I have the sketch below from Desmos, and I hope my point is getting across.
Can somebody shed light on how to handle such situations, in general? What is the process to find the volume of solids generated, when the axis of rotation intersects the region which is under consideration, particularly when the stretch of both functions on either side of this line are so close? I am looking for both a particular solution to the problem above, and a generic guidance. Any clues, please!!
Graphically, it is relatively simple. When the region has rotated $\pi$ radians around the line $y=1,$ it will be back in the $x,y$ plane but a mirror image of the original region, reflected through the line $y=1.$
The mirror-image region is still bounded by a line and a parabola, but the line and parabola also are mirror images of the original line and parabola, reflected through the line $y=1.$
Shade the original region and shade the reflected region. The total shaded region (union of the original and reflected regions) is a cross-section (in the $x,y$ plane) of the solid of revolution. The slices of this solid perpendicular to its axis are either disks or washers bounded by one or two bounding functions (a line and/or a parabola). You can tell which function(s) to use on which interval of $x$ values by looking at the intersections of the functions with each other and with the axis of rotation. Note that you will likely have to do several separate integrals.