Help with Green's theorem

• May 13th 2010, 05:37 PM
thunderdan7000
Help with Green's theorem
<b>The Problem:</b>
a) Use Green's Theorem to calculate ∫F*dr over C, where F(x,y)=(2e^x-3y^2+5x)i+(xy-2x+ysiny)j and C is a piecewise smooth curve defined as C1: y=0, -2 ≤x ≤-1; C2: arc of x^2+y^2=1 from Θ= ∏/4 to ∏; C3: Θ=∏/4, r=1 to 2; and C4; the arc of x^2+y^2=4 which connects the other 3 sides. The curve starts on C1 at (-2,0) and is traversed in a counterclockwise direction.
b) Get the same answer by computing the line integral for work.

<b>Green's Theorem</b>
∫F*dr = Q/ x- P/ y dA where P is the x component and Q is the Y component of the given function F.

<b>What I've got</b>
Not a whole lot; I have:
Q/ x=y-2 and P/ y=6y. I understand that I am supposed to treat each side independently, but I can't figure out how to set up or evaluate the integrals. For example on C1 y=0 and this makes my entire integral go to zero. This may be the correct answer for that side but I'm not at all sure if I'm even evaluating it correctly. Any pointers as to how to set up the integrals would be tremendously appreciated, Thank you all.

• May 13th 2010, 09:32 PM
matheagle
you have C2 wrong
It's from $\pi$ to $\pi/4$

In any case, just use polar where you integrate $1\le r\le 2$ and $\pi/4 \le \theta \le \pi$
Since it's counterclockwise the sign is positive and do not forget the Jacobian of r.