Math Help - Apply Green's Theorem to evaluate the integral

1. Apply Green's Theorem to evaluate the integral

$\displaystyle \oint_{C}4ydx+2ydy$

Where $C: 0\leq x\leq \pi$, $0\leq y \leq sin(x)$

Ok, so I let $M=4y$ and $N=2y$

And then

$\displaystyle \iint_{R} \frac{\partial N}{\partial x}-\frac{\partial M}{\partial Y}dxdy$

Comes out to

$\displaystyle \int_{0}^{\pi}\int_{0}^{sin(x)} -4dydx$

Then

$\displaystyle \int_{0}^{\pi}[-4y|^{sin(x)}_{0}]dx$

Which gives me

$\displaystyle \int_{0}^{\pi}-4sin(x)dx$

$\displaystyle 4cos(x)|^{\pi}_{0}$

Which is

$-4-4=-8$

The answer in the book, however, is -4. What am I doing wrong? Thanks

2. Originally Posted by downthesun01
The answer in the book, however, is -4. What am I doing wrong? Thanks
Nothing, your book is wrong. The answer is $-8$ .

Regards.

Fernando Revilla