Find the outward flux of the field - Green's Theorem

[mit_hacker]

Homework Statement

(Q) Find the outward flux of the field

F=(3xy-x/(1+y^2 ))i+(e^x+tan^(-1)⁡y )j

across the cardioid r=a(1+cos⁡θ), a>0.

Homework Equations

div F = (∂M )/∂x+∂N/∂y

The Attempt at a Solution

I could easily set up the double integral which is:

∬▒3 r^2 sin⁡θ dA

However, I am unsure as to how to determine the limits to be used.
Please help. Thank-you.

 

[HallsofIvy]

?? The limits of integration ARE the cardioid, of course: \theta going from 0 to 2\pi, r from 0 to a(1+ cos(\theta)). Since that is given in polar coordinates, it might be best to convert div F to polar coordinates.

 

[mit_hacker]

Yes but...

Usually, the limits of r ranges from a constant value to another constant value of r. Why do we, in this case simply plug in the equation of the cardioid?

Thanks a lot for your help and support!