So what they did was transform to a coordinate system that is easier to integrate.

You can just plug in values to see what happens to x as t varies, it shows that as x varies 0->a, t varies 0->Pi/2.

We know , so

, this is our Jacobian.

Then we have:

Area =

But we have cos(t)^2 = 1 - sin(t)^2 which gives:

I'm not 100% sure why their limits are Pi/2 to 2 instead of my 0 to Pi/2 but it doesn't seem to change the answer.