Let R be the rectangle 0 ≤ x ≤ a, 0 ≤ y, ≤ b. Evaluate ∫_{R} e^{f(x, y)} dx dy, where f(x, y) = max(b^{2}x^{2}, a^{2}y^{2}).

**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) = b^{2}x^{2}. 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 = a^{2}b^{2}. So the required value is twice this or (e^{k} - 1)/(ab).

© John Scholes

jscholes@kalva.demon.co.uk

14 Aug 2002