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Thread: Apply Green's Theorem to evaluate the integral

  1. #1
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    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
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by downthesun01 View Post
    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
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