Use Green's Theorem to evaluate the line integral along the positively oriented closed curve $C$.
$\oint _ { C } ( 6 x y + 5 \ln ( 1 + x ) ) d x + 2 x ^ { 2 } d y$, where $C$ is the cardioid $r = 2 + 2 \cos \theta$.
A) $- 20 \pi$
B) $- 80 \pi$
C) $- 10 \pi$
D) $- 40 \pi$

Question

Quizplus · Accepted Answer

Using Green's Theorem, the line integral becomes$$\begin{split} \oint_C (6xy+5\ln(1+x))dx+2x^2dy &=\iint_R \left( \frac{\partial}{\partial x}[2x^2]-\frac{\partial}{\partial y}[6xy+5\ln(1+x)] ight) dA \\ &=\iint_R (4x-6x) dA \\ &=0. \end{split}$$

Use Green's Theorem to Evaluate the Line Integral Along the Positively