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Green's theorem

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Apply Green's Theorem to evaluate the integral over C of 2(x^2+y^2)dx + (x+y)^2 dy, where C is the boundary of the triangle with vertices (1,1), (2,2) and (1,3) oriented in the counterclockwise direction.

Also check the result by direct integration.

Please show detailed working so I can follow the steps of the working.

Thank you

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Solution Summary

This shows how to use Green's theorem to evaluate an integral. The triangle vertices oriented in the counterclockwise directions are given.

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Please see the attached doc files

First we use the Green's Theorem in the plane:
The closed surface is:

We consider:

Then :

which is:

on the given surface. We can divide ...

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