Homework Help: Problems with Green's Theorem

1. Nov 23, 2004

Chronocidal Guy

K, I'm puzzled to death on a two problems involving Green's Theorem. They both are asking me to confirm that Green's theorem works for a given example, so I have to compute both the double integral over the area and the integral over the closed curve and make sure that they match.. only, on one problem the answer's don't match at all, and the other I'm stuck setting up the integral.

First one: Integrate the function F= x*yi +x^2*yj over a rectangle that has points (0,0), (a,0), (a,b), (0,b), going counterclockwise. I've gotten the double integral for the area perfectly according to the answer section ( (a*b^2)/2 + (a^3*b)/3 ), but the answer I get for the curve integral doesn't match. I broke the square into four curves, C(1) through C(4), and integrated each separately using F (dot) n, the dot product of the function and the exterior normal vector. But, when I do this, I get four separate integrals, which form two pairs of identical integrals in opposite directions.. they all cancel and the answer I get is zero. The answer guide for my textbook does it the same way, but the two negative terms that cause the cancellation just seem to be dropped during the last step for some reason. Am I missing something really simple here that lets those two integrals drop away?

Second: Nearly identical problem, except in this case, F= cos(x+y)i + sin(x+y)j, and the area in question is a triangle with points (0,0), (a,0), and (a,b). Again, the double integral for the area went smoothly, but I'm stuck setting up the three separate curve integrals. I've gotten the two legs of the triangle, but how do I set up the integral along the hypoteneuse? I'm guessing I need a double integral since for the third side x is going from a to zero, and y is going from b to zero... but the equation I need to integrate is F (dot) n = cos(x+y)dy/ds + sin(x+y)dx/ds . Do I need to evaluate one integral, and then plug the answer from that into the next integral, or do I need to add them together, or do I need to do something entirely different?

Hope this makes sense to whoever reads this, I'm just kind of panicking at the moment because I have a physics midterm tomorrow, and the math homework is ruining my study time. :P

2. Nov 24, 2004

Galileo

Be sure to get the right values for x and y when traversing the different line segments. i.e. in going from (0,0) to (a,0), y=0 all the way. On the way back however from (a,b) to (0,b), y=b along the way. Don't treat y as a variable which cancels out at the end. The integral along any line segment should be just a number.