Compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for the oriented curve specified.

\(\mathbf{F}(x, y)=\left\langle e^{y-x}, e^{2 x}ightangle\), piecewise linear path from \((1,1)\) to \((2,2)\) to \((0,2)\)