How to Verify Green's Theorem for a Given Rectangle?

[sam121]

Homework Statement

Verify Green's Theorem in the plane for the \oint [(x^{2} - xy^{2})dx + (y^{3} + 2xy)dy] where C is a rectangle with vertices at (-1,-2), (1,-2), (1,1) and (-1,1).

The Attempt at a Solution

This means you have to use green's theorem to convert it into a double integral and solve which I have done. Not 100% sure whether the answer is 6 or 12 however. You then have to do the line integral directly to verify you get the same answer. This is where I get stuck. Please could I have some help on how to do this line integral directly? thank you :)

 

[jackarms]

You can do it a couple different ways:

One is to parameterize each segment of the rectangle and evaluate four integrals in terms of t. For example, the path from (-1, -2) to (1, -2) can be parameterized as \stackrel{\rightarrow}{r}(t) = \left\langle -1 + 2t, -2\right\rangle where 0 \leq t \leq 1, so you can have x = -1 + 2t, y = -2, dx = 2dt, and dy = 0dt. Then just substitute everything and evaluate, then repeat the process three more times.

Another way, since all the paths are straight lines either vertical or horizontal, is to evaluate the integrals without parametizing the path. Make a specific path of vertices, say the path you listed of (-1,-2) \rightarrow (1,-2) \rightarrow (1,1) \rightarrow (-1,1) \rightarrow (-1, -2), then create integrals from each of those paths. For example, for the first path x goes from -1 to 1 and y = -2 and does not change, so the integral becomes:
\int^{1}_{-1}x^{2} - 4xdx.

Either way will work.