Calc Q: Line Int. Depends on Area, Not Placement

[Sneaksuit]

Anyone know how to do this question?
Let C be the boundary of any rectangular region in R2. Show that the value of the line integral
$$\oint (x^2 y^3 -3y)dx + x^3 y^2 dy$$
depends only on the area of the rectangle and not on its placement in R2.

 

[Data]

Can you think of a good way to start the question (I'm assuming you've learned a few useful theorems on line integrals)?

BIG HINT: What theorem will let you incorporate the area of the rectangle into the evaluation of the integral?

 

[Sneaksuit]

Well, we are working on Green's Theorem right now but i don't remember it incorporating the area of a rectangle. So, to answer your question...no, i don't even know where to begin.

 

[arildno]

You must parametrize your boundary:
Let "t" be in some interval, so that the perimeter of the rectangle is given as some path (x(t),y(t)). Note that $$dx=\frac{dx}{dt}dt$$
similarly for dy, and that on regions where say x=constant, dx must equal 0.

 

[Crosson]

Green's theorem applies to rectangles, and any curves that have a finite number of corners and don't cross themselves.

 

[arildno]

Green's theorem applies to rectangles, and any curves that have a finite number of corners and don't cross themselves.

I din't imply that you couldn't use Green's theorem.
Of course you can, and it is probably the easiest way to do this.

(I thought to be "creative" in giving an alternative way of doing this, but reviewing the problem, following my earlier "advice" is simply inadvisable..)

 

[Data]

Green's Theorem is definitely the way to go for this problem. Just apply it and see where it leads you.