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Math Help - Another line integral question....

  1. #1
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    Another line integral question....

    Evaluate the line integral directly, WITHOUT using Green's theorem.

    Integral of (x-y)dx+(x+y)dy where C is the circle with center the origin and radius 2.

    I have to show that both methods give me the same answer. I know how to do it with Green's theorem...but not otherwise.
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  2. #2
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    Change to polar coordinates

    Rectangular and polar coordinates have the relation:
    <br />
x = r cos \theta<br />
    <br />
y = r sin \theta<br />
    from which we get:
    <br />
dx = cos \theta dr - r sin\theta d\theta<br />
    <br />
dy = sin \theta dr + r cos\theta d\theta<br />
    Substitute into your problem:
    (x-y)dx = (rcos\theta - rsin\theta)(cos\theta dr - r sin\theta d\theta )<br />
    (x+y)dy = (rcos\theta + rsin\theta)(sin\theta dr + r cos\theta d\theta )<br />
    Your contour is a circle, so r=constant, so the integral over r is 0 - we can ignore the dr terms.
    Multiply out the rest to get to get
    <br />
(x-y)dx + (x+y)dy=<br />
-r^2 sin \theta cos \theta d \theta + r^2 sin^2 \theta d\theta + <br />
r^2 cos^2 \theta d\theta + r^2 sin \theta cos\theta d\theta<br />
    The RHS collapses to r^2 d\theta
    Your integral is for r = 2, and \theta going from 0 to  2 \pi
    so it equals 2^2 * 2  \pi = 8 \pi
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