Evil integration problem!

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Evil integration problem!

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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

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|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?

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