volume under a 3-d curve

Let $a,b > 0.$ Find $int_R e^{f(x,y)}dxdy,$ where $f(x,y) = max{a^2y^2, b^2x^2}$ and $R = {(x,y) : 0leq x leq a, 0leq y leq b}.$

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 $2int_0^a int_0^{frac{b}a x} e^{b^2x^2} dydx = dfrac{e^{a^2 b^2}-1}{ab}.$