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Thread: Line Integral

  1. #1
    Member Em Yeu Anh's Avatar
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    Red face Line Integral

    Evaluate $\displaystyle \int_Csinxdx+cosydy$ where C is the top half of the circle $\displaystyle x^2+y^2=4$ from (2,0) to (-2,0) and the line segment from (-2,0) to (-3,2).

    Haven't done vector calc in awhile, still a little bit rusty, can someone assist me with getting the parametric equations for the line segment?
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    MHF Contributor Bruno J.'s Avatar
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    The line segment from $\displaystyle p_1$ to $\displaystyle p_2$ can be parametrized as $\displaystyle r(t)=(1-t)p_1+tp_2$, $\displaystyle 0 \leq t \leq 1$.
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    Super Member 11rdc11's Avatar
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    Quote Originally Posted by Em Yeu Anh View Post
    Evaluate $\displaystyle \int_Csinxdx+cosydy$ where C is the top half of the circle $\displaystyle x^2+y^2=4$ from (2,0) to (-2,0) and the line segment from (-2,0) to (-3,2).

    Haven't done vector calc in awhile, still a little bit rusty, can someone assist me with getting the parametric equations for the line segment?

    For the circle let

    $\displaystyle x=2\cos{t}$ and $\displaystyle dx = -2\sin{t}$

    $\displaystyle y =2\sin{t}$ and $\displaystyle dy = 2\cos{t}$

    Now sub these values for

    $\displaystyle \int_Csinxdx+cosydy$

    and you get

    $\displaystyle C_1 = \int^{\pi}_{0} \bigg(-2\sin{t}\sin{(2\cos{t})}+ 2\cos{t}\cos{(2\sin{t})}\bigg)dt$

    For $\displaystyle C_2$

    $\displaystyle x = -2 -t$ and $\displaystyle dx= -1$

    $\displaystyle y =2t$ and $\displaystyle dy = 2$

    and then just sub in like before.
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