50th Putnam 1989

Let R be the rectangle 0 ≤ x ≤ a, 0 ≤ y, ≤ b. Evaluate ∫_R e^{f(x, y)} dx dy, where f(x, y) = max(b²x², a²y²).

Solution

Divide the rectangle into two by the diagonal from (0, 0) to (a, b). The required value is evidently twice the integral over the lower half of e^g(x), where g(x) = b²x². But integrating over y just gives a factor bx/a, and the integration over x is then trivial to give (e^k - 1)/(2ab), where k = a²b². So the required value is twice this or (e^k - 1)/(ab).

50th Putnam 1989

© John Scholes
jscholes@kalva.demon.co.uk
14 Aug 2002