Evil integration problem!

Question

anonymous · Answer

Concerning the first law of gravitation, I am seeing whether the same result can be achieved by integration as if you treat a planet (orbiting the son) as a point-like particle (at its centre of mass)

anonymous · Answer

Firstly, consider the sun. If we draw a circle around the it, we can say that the same force will act on 1kg anywhere in the circle.|dw:1328041371179:dw|

anonymous · Answer

Now, consider a planet orbiting the sun|dw:1328041409993:dw| (Not to scale)

anonymous · Answer

|dw:1328041434214:dw|
All the 'particles' in the arc will experience the same force.

The area of the arc is, for any point x-t :$$2\pi*q*\theta/360$$

All terms in the drawing can be easily worked out as functions of x, taken that z and r are constants. 

What would you actually do to work out the area and the force acting on it due to Newton's First Law of Gravitation by integration?