Let $C_{1}$ be the curve y = x as x goes from 0 to 1, and $C_{2}$ be the curve y = sin($\frac{\pi*x}{2})$ as x goes from 0 to 1. Which is larger,
$\int_{C_{1}} e^{y^{2}}dx + e^{x^{2}}dy\:\:\:$ or $\:\:\:\int_{C_{2}} e^{y^{2}}dx + e^{x^{2}}dy?$
Letting $C$ be the combination of $C_{1}$ and $C_{2}$ (with positive orientation), $C$ is a closed surface. So I figured that the sign of the integral $\int_{C} e^{y^{2}}dx + e^{x^{2}}dy\:\:$ would tell me "which is larger". However, I am having trouble determining the sign of the integral. Solving the integral exactly (even with Greene's Theorem) seems out of the question because of the integral's complexity.
So how do I evaluate the integral? Is my general approach correct?
Your approach will work. Via Green's theorem you want to integrate $2xe^{x^2}-2ye^{y^2}$ over the interior of the region bounded by $C$. Note that this region is contained in the half plane $y \geq x$. Also, the function $2xe^{x^2}$ is a strictly increasing function. So the integrand $2xe^{x^2}-2ye^{y^2}$ is going to be negative at points inside $C$, and so the double integral will be negative. And so since $\partial C=C_1-C_2$, the integral over $C_2$ will be larger.