[bizaro86]

Another thought that is still not a solution but maybe helps someone?

Spoiler!

I mean, the remaining piece is even so you could re-write as

integral[sqrt(4-x^2)] (from 0 to 2)

That doesn't help me get any closer to an actual solution though, because I still don't know how to evaluate that integral.

I do still think trig substitutions could work, but I don't remember how to do that so I couldn't move forward without a reference which doesn't seem in the spirit of this.