Problem: Evaluate Integral F dot dr, 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. F(x,y)=(e^(-x)+3y)i+(x)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 2. Relevant equations Integral F dot dr=DOuble integral over the region R, (dg/dx-df/dy)dA 3. The attempt at a solution I started by completing the square of that circle that is not centered at the origin, and got (x-1)^2+y^2=4. So now I know the inner region's boundary is a circle of radius 2 centered at (1,0). Also, I got the double integral over x^2+y^2=16 , Double integral 0to 2pi, 0 to 4 (-2)r dr dtheta and got -32pi. But I don't know what to do from here. The circle x^2-2x+y^2=3 is giving me a hard time. Can you tell me how to do the rest of this problem?