Chapter 15.4, Problem 40ES

Evaluate the integral ${\displaystyle {\int }_C\text{F}\cdot d\text{r},}$ where C is the boundary of the region R and C is oriented so that the region is on the left when the boundary is traversed in the direction of its orientation.

$\text{F}\left(x,y\right)=\left(e^{-x}+3y\right)\text{i}+x\text{j};C$ is the boundary of the region R inside the circle $x^2+y^2=16$ and outside the circle $x^2-2x+y^2=3.$