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Thread: 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
    Senior Member
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    Change to polar coordinates

    Rectangular and polar coordinates have the relation:
    $\displaystyle
    x = r cos \theta
    $
    $\displaystyle
    y = r sin \theta
    $
    from which we get:
    $\displaystyle
    dx = cos \theta dr - r sin\theta d\theta
    $
    $\displaystyle
    dy = sin \theta dr + r cos\theta d\theta
    $
    Substitute into your problem:
    $\displaystyle (x-y)dx = (rcos\theta - rsin\theta)(cos\theta dr - r sin\theta d\theta )
    $
    $\displaystyle (x+y)dy = (rcos\theta + rsin\theta)(sin\theta dr + r cos\theta d\theta )
    $
    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
    $\displaystyle
    (x-y)dx + (x+y)dy=
    -r^2 sin \theta cos \theta d \theta + r^2 sin^2 \theta d\theta +
    r^2 cos^2 \theta d\theta + r^2 sin \theta cos\theta d\theta
    $
    The RHS collapses to $\displaystyle r^2 d\theta$
    Your integral is for r = 2, and $\displaystyle \theta $ going from 0 to $\displaystyle 2 \pi$
    so it equals $\displaystyle 2^2 * 2 \pi = 8 \pi$
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