You're looking for something fun for a calculus course? If a rectangle $R$ is tiled by rectangles, each of which has a side with integer length, then $R$ has a side with integer length. This is from
Wagon, Stan. Fourteen proofs of a result about tiling a rectangle. Amer. Math. Monthly 94 (1987), no. 7, 601--617. MR935845
and one of those fourteen proofs goes by a double integral.