Concept Check for two Green Theorems Problems

[stratusfactio]

Homework Statement

**The book I'm working from doesn't have solutions for even numbered problems and the two problems I'm about to show you guys are even, so if you guys don't mind, I would just like to know if I have the right concept, or if I messed up anywhere. I wouldn't have a way of knowing since there aren't answers for the two problems in the back of the book.**

For the following problems use Green’s Theorem to evaluate the given line integral around the
curve C, traversed counterclockwise.

1. Closed integral of C(e^(x^2)+ y^2) dx + (e^(y^2)+ x^2) dy; C is the boundary of the triangle with vertices (0,0),(4, 0) and (0, 4)

2. Evaluate closed integral of C (e^x sin y) dx + (y^3 + e^x cos y) dy, where C is the boundary of the rectangle with vertices (1, −1), (1, 1), (−1, 1) and (−1, −1), traversed counterclockwise.

Homework Equations

The Attempt at a Solution

[1] I got 0 for the answer. We see that P = e^(x^2)+ y^2 and Q = e^(y^2)+ x^2, so taking derivative of Q w/ respect to x we obtain 2x and taking derivative of P w/ respect to y we obtain 2y. Therefore we have the double integral w/ both integrands ranging from [0,4] of 2x - 2y.

[2] I got 0 for the answer here also. We see that P = e^x sin y and Q = y^3 + e^x cos y, so taking derivative of Q w/ respect to x we obtain e^x cos y and taking derivative of P w/ respect to y we obtain e^x cos y as well. Therefore we have the double integral w/ both integrands ranging from [-1,1] of e^x cos y - e^x cos y ( or 0 and therefore answer is 0).

 

[ArcanaNoir]

Your answers are correct but in the first problem your int dx should run from 0 to 4-y. It came out the same this time, however.

 

[stratusfactio]

Ahhh, I see. Is that because the diagonal of the triangle is the line 4-y?