Well this disturbs me.

We have two intersecting curves

and

We want the area between the curves.

Now, according to my text the area contained only within the cosine curve will be

So then we subtract this from the area of the cosine curve.

The student was insisting that the answer is this:

These give different answers.

I know that the text is right (if I set it up properly, that is), but the region of integration of the cosine circle doesn't, in fact, start with the angle 0.

It's been a long time since I've done these, folks. Please forgive me if I'm missing the obvious.

-Dan