Thread: Another line integral question....

1. 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.

2. Change to polar coordinates

Rectangular and polar coordinates have the relation:
$
x = r cos \theta
$

$
y = r sin \theta
$

from which we get:
$
dx = cos \theta dr - r sin\theta d\theta
$

$
dy = sin \theta dr + r cos\theta d\theta
$

Substitute into your problem:
$(x-y)dx = (rcos\theta - rsin\theta)(cos\theta dr - r sin\theta d\theta )
$

$(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
$
(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 $r^2 d\theta$
Your integral is for r = 2, and $\theta$ going from 0 to $2 \pi$
so it equals $2^2 * 2 \pi = 8 \pi$