I have no idea how to solve the double integral for the function. Could I get some help please?
$\displaystyle
\int_0^1 \int_0^1 max\{x,y\} * e^{max\{x^2,y^2\}} dx dy
$
Here's a non-rigorous (but essentially correct) way to tackle this. For 0 ≤ t ≤ 1, the set $\displaystyle \{(x,y):\max\{x,y\}=t\}$ is L-shaped, consisting of two line segments each of length t. Thus it has "infinitesimal area" 2tdt, and therefore $\displaystyle \int_0^1 \!\!\int_0^1\!\! \max\{x,y\} * e^{\max\{x^2,y^2\}}\, dx\, dy = \int_0^1\!\!te^{t^2}2t\,dt$. You can integrate this by parts, but the answer will have to involve the error function. Numerically, I make it approx. 1.9715.