The washer method is just a special case of taking slices.
Slice the torus perpendicular to the x-y plane, going from y=-a to y=a.
Each slice is an annulus = a disk with an inner disk removed.
At height , the outer radius is the distance from to , where is on the original circle with .
At height , the inner radius is the distance from to , where is on the original circle with .
By symmetry, it's obvious that .
Thus Outer Radius = . Inner Radius = .
Area(y) = pi (Outer Radius)^2 - pi (Inner Radius)^2.
But on the circle, , so .
Now when when .
(Note on this interval.)