I just thought about this idea and I decided to work on it.
After taking on a general case, which proved to be too difficult, I tried a specific case. Something simple like the curve $y_1 = x^2$ rotating about the line $y = x$
Which is the same as rotating $y = \sqrt{x}$ about the x-axis.
I know I need to find the new radius which is the line perpendicular to y = x and I need to pick a particular point on the curve and the line.
So if i were to pick say, x = 0.5, the perpendicular line would be
$y =-x + 1$
So my solid of revolution integration would be
$\pi \int_{a}^{b} (-x + 1)^2 d?$
Unfortunately it proved to be very difficult to find the slanted differential in terms of dx and I couldn't figure out what the change of variables of bounds were.
Any ideas?