Chapter 15.4, Problem 46E

Green’s Theorem: Region with a Hole

Let R be the region inside the ellipse $x=4\cos \theta$, $y=3\sin \theta$ and outside the circle $x=5\cos \theta$, $y=5\sin \theta$ and outside the circle $x=2\cos \theta$, $y=2\sin \theta$. Evaluate the line integral

${\displaystyle \underset{C}{\int }(3x^{2}y+1)dx+}(x^3+4x)dy$

where $C=C_1+C_2$ is the boundary of R, as shown in the figure.