I am working on this problem and I can't quite figure it out. For P I get x^3 because that's multiplied by dx. And for Q I get 5x. So dq/dx-dp/dy = 5.
Plugging that in I get double integral over D of 5 dA. Next I realize that there to do dydx as my dA I have to split the parallelogram into two pieces. So I'm integrating two separate triangles. My guess is I'm doing that wrong but I can't figure out why. I decided to do the line integral by finding four vectors going all the way around the shape by that just got me a completely different answer.
What am I doing wrong?
Thank you,
Hello sir. I keep trying to do this problem as you described on your blog,
but I keep reaching the same answer I got last time. Does this mean I'm
still doing it wrong or that the webwork is wrong??
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It's hard to see what you did wrong without having the details of your calculation. First of all note that the orientation of the contour is clockwise, while Green's theorem is framed for counterclockwise line integrals. So: just go ahead and use Green's theorem as normal and then multiply your answer by -1. Incidentally, I would do the double integral as an iterated integral with dx first, since that way I can do only one piece. The leftmost sloping line segment runs from (0,0) to (x_0,y_0) so that line can be written as y=y_0/x_0 x or (as you will need it) x=x_0/y_0 y. The rightmost sloping line segment is displaced from the first by an amount x_0 so has equation x=x_0/y_0 y + x_0, so the iterated integral is
which is pretty easy to do. (Updated)Since here x_0=2 and y_0=2, the inner integral is 5((2/2)y+2-(2/2)y)=10. Then you get the integral of 10 with respect to dy from 0 to 2, which is positive 20. The negative of that is -20, which is the answer you got so I believe webwork is failing to properly check your answer.
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