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/int_C (xdy-ydx) / (x^2 + y^2 + 1)
where C is the boundary of the minor segment of the circle x^2 + y^2 = 1 cut off by the chord x+y=1
I've sketched the graph and can see the segment needed but really don't know where to go with this one as i've not done one like this before.
Very fustrated!!!
Which way are you going with that? How about in the positive sense. Then we'd have:
$\displaystyle \mathop\oint\limits_{C}\frac{xdy}{x^2+y^2+1}-\frac{dx}{x^2+y^2+1}$
On the arc, parameterize it as $\displaystyle x=\cos(t),\; y=\sin(t)$ and make that substitution into the integral (dx=-sin(t)dt and dy=cos(t)dt and the limits go from 0 to pi/2 right?). Now parameterize the chord as $\displaystyle x=t,\; y=1-t$ with the limits going from 0 to 1. Now make that substitution.
