Year: 1989
Topic: Calculus

Problem:
Evaluate $\displaystyle{\int_0^a\int_0^b e^{{\rm max}\{b^2x^2, a^2y^2\}}\,dy\,dx}$ where $a$ and $b$ are positive.

Solution:
Divide the region into two parts by the diagonal line $ay = bx$ to get 
\[ 
\int_0^a\int_0^b e^{{\rm max}\{b^2x^2, a^2y^2\}}\,dy\,dx = \int_0^a \int_0^{bx/a} e^{b^2x^2}\,dy\,dx + \int_0^b \int_0^{ay/b} e^{a^2y^2}\,dx\,dy 
\]
\[ 
= \int_0^a \frac{bx}{a} e^{b^2x^2}\,dx + \int_0^b \frac{ay}{b} e^{a^2y^2}\,dy 
\]
\[ 
= \boxed{\frac{e^{a^2b^2} - 1}{ab}}. 
\]