Using Green's Theorem for a quadrilateral

[Mohamed Abdul]

Homework Statement

Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful.

Homework Equations

Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given here: http://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx

The Attempt at a Solution

In order to solve this problem, I thought I could just utilize Green's theorem at the bounds of the shape. My main question, however, is what my TA told me: apparently in order to evaluate the integral I also have to subtract the line integral over the curve from (0,6) to (0,0). I am very confused on why we need to do that.

 

[Ray Vickson]

Homework Statement

Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful.

Homework Equations

Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given here: http://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx

The Attempt at a Solution

In order to solve this problem, I thought I could just utilize Green's theorem at the bounds of the shape. My main question, however, is what my TA told me: apparently in order to evaluate the integral I also have to subtract the line integral over the curve from (0,6) to (0,0). I am very confused on why we need to do that.

As stated, the question did not say "closed path", so it goes from A to B to C to D, but not from D back to A. If you used Green's theorem, you will have included a line-segment that was not part of the original problem.

 

[Mohamed Abdul]

As stated, the question did not say "closed path", so it goes from A to B to C to D, but not from D back to A. If you used Green's theorem, you will have included a line-segment that was not part of the original problem.

So in solving this I'd use Green's Theorem as normal and then simply subtract the line integral for the path from D to A?

 

[LCKurtz]

So in solving this I'd use Green's Theorem as normal and then simply subtract the line integral for the path from D to A?

Yes. Not sure why you call the shape a parallelogram in post #1. Also, the integrand in Green's theorem comes out constant. Does that suggest a shortcut?