Each of m and n is a positive integer with m < n.

Evaluate this double definite integral in terms of m and n.

∫ ∫ {√x + √y } dx dy, for x = 1 to m² and, y = 1 to n²

Note: {u} = u - [u], where [u] denotes the greatest integer ≤ u

*** For an extra challenge, solve this puzzle without the aid of a computer program or, an online integrator.