For additional integration practice, here is a problem:

Given two randomly selected points in a unit square, what is the probability that the distance between them is greater than 1/2?

(2015 AMC A, modified)

The original problem specified that the points lie on the perimeter of the square, which makes the problem significantly easier, and allows you to use geometry rather than calculus.

Edit: Also, sqrt(1 - x^2) would be one of the integrals you would need to evaluate in the original problem.