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Math Help - line integral

  1. #1
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    line integral

    I want to compute the line integral

    integral (2ydy-ydx) where C is the boundary of the half disc (x^2)+(y^2) less than or equal to 1 with y greater than or equal to 0 traversed in the positive sense by parametrising the boundary curve (there are two pieces, a straight line segment and a semi-circle) and then evaluating the integral directly.

    I did this:
    integral (2ydy-ydx)
    I use the following:
    integral (f1(x,y)dx+f2(x,y)dy)= double integral (df2/dx-df1/dy)dA

    Thus, I have I=double integral(d/dx(2y)+d/dy(y))dA
    I=double integral (1)dA=integral (from 0 to pi)dtheta integral (from 0 to 1) 1dr=integral (from 0 to pi) 1dtheta=pi.

    I got an answer of pi for this question, but it seems too easy. Is this correct?

    I used Green's Theorem by the way.
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  2. #2
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    You need to find,
    \oint_C -ydx+2ydy
    Where C is a positively oriented, piecewise smooth, closed simply connected region.
    Thus, we can use Green's theorem.
    \oint_C -ydx+2ydy=\int _D\int \left( \frac{\partial (2y)}{\partial x}-\frac{\partial (-y)}{\partial y}\right) dA=\int_D\int dA
    Thus it is simply the area of the region.
    In this case,
    \frac{1}{2}\pi
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