# 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:
$\displaystyle x = r cos \theta$
$\displaystyle y = r sin \theta$
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 $\displaystyle 2^2 * 2 \pi = 8 \pi$