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 .

Thus .

Thus .

Now when when .

(Note on this interval.)