Hey Lana K.,
Don't get thrown off by the unusual format of the "function" (yes, I know, it is a relation only). What I would do, just reverse all the variable references, and solve like a normal rotation around the y-axis. That's easier to visualize, for sure. That (the latter) is a downwards-pointing parabola (as a cross section on the x-y plane, then rotated around the y-axis), with x-axis intercepts of 0 and 2 and a maximum [vertex] at (1,1). Slices of that (perpendicular to the y-axis) will be described as circle-annuli, with an inner and outer radius given by the x-values of the parabola at those particular y-values. You know how to express the areas of those two limit circles; the area of each annular ring is the outer-circle area minus the inner-circle area, isn't it.
Now comes the catch: how do you force the values for the inner and outer circles? It wouldn't be easy in the expressed equation format. So, reach into your grab-bag of "tricks", and solve as a simple parabola with Cartesian "axes" relocated to the parabola vertex: (1-y) = (x-1)^2 . That form is easier to solve through for x, isn't it: (1-y)^0.5 = ±(x-1) for the two parabola lobes. Now you can make progess at expressing your two values for x (radii of the two circles). That is, I hope you can!
--Cheers, --Mr. d.
