Let $a,b > 0.$ Find $\int_R e^{f(x,y)}dxdy,$ where $f(x,y) = \max\{a^2y^2, b^2x^2\}$ and $R = \{(x,y) : 0\leq x \leq a, 0\leq y \leq b\}.$
I think I should consider symmetry over the rectangle $R$ (perhaps dividing the rectangle in half might be useful). I need to determine a way to "isolate" $b^2x^2$ or $a^2x^2$ from $f(x,y)$ to make it easier to integrate. I tried playing around with specific values (e.g. $a=b=1$), but I'm not sure how to derive the general pattern.
Edit: I think I finally came up with a solution to this problem, shown below. Let me know what you think.
So basically, as pointed out, it's useful to consider the diagonal of the rectangle, which has equation $y = \dfrac{b}a x.$ Clearly, the volume is twice the volume over the lower half where $y \leq \dfrac{b}a x.$ Since $a^2y^2 \leq b^2x^2$ over this region, the volume is $2\int_0^a \int_0^{\frac{b}a x} e^{b^2x^2} dydx = \dfrac{e^{a^2 b^2}-1}{ab}.$