Rayan's question at Yahoo Answers (Green's Theorem)

[Fernando Revilla]

Here is the question:

Use Green's Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.
F(x,y) = (e^x − 9y)i + (e^y + 4x)j
C: r = 2 cos(θ)

Here is a link to the question:

Use Green's Theorem to calculate the work done by the force F? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Rayan,

Easily proved, $C:r=2\cos \theta$ is the circle $C:(x-1)^2+y^2=1$. If $D$ is the disk with boundary $C$, then by the Green's theorem, $$W=\int_C(e^x − 9y)dx + (e^y + 4x)dy=\iint_D(Q_x-P_y)dxdy=\iint_D(4+9)dxdy\\=13\iint_Ddxdy=13\mbox{Area }(D)=13\cdot \pi\cdot 1^2=\boxed{\;13\pi\;}$$
If you have further questions you can post them in the http://www.mathhelpboards.com/f10/ section.http://www.mathhelpboards.com/f10/