Kaoyan 2025 Math II, Problem 20 | A Double Integral over Two Overlapping Disks

Compute ∬_D (x−y)² dx dy over the overlap of the discs x²+y²≤4x and x²+y²≤4y. We start with what a double integral adds up (cut into pieces, value times area, add them all, then in two rounds as an iterated integral), which is why step one must be drawing the region. Four steps: draw D, set up the integral, compute it, check. Completing the square gives two discs of radius 2 whose overlap is a leaf with tips at (0,0) and (2,2). In x and y the limits carry square roots and the algebra turns into cubes, so we go polar: both circles pass through the origin and become r ≤ 4 cos θ and r ≤ 4 sin θ, with dA = r dr dθ. Along each ray the first circle hit wins, so the upper limit switches from 4 sin θ to 4 cos θ at θ = π/4. The fold across y = x lets us integrate the lower half and double it; power reduction and a substitution give 12π − 112/3 ≈ 0.37. To check, the distance to the diagonal gives the estimate 0 < I < 1.57, a thin-trough volume explains why the answer is so small, and the two classic boundary mistakes, two smaller traps and the exam write-up close the lesson.