# Apply Green's Theorem to evaluate the integral

• Dec 8th 2010, 10:25 PM
downthesun01
Apply Green's Theorem to evaluate the integral
$\displaystyle \displaystyle \oint_{C}4ydx+2ydy$

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

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

And then

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

Comes out to

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

Then

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

Which gives me

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

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

Which is

$\displaystyle -4-4=-8$

The answer in the book, however, is -4. What am I doing wrong? Thanks
• Dec 8th 2010, 11:55 PM
FernandoRevilla
Quote:

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 $\displaystyle -8$ .

Regards.

Fernando Revilla