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October 6th, 2017, 12:50 AM   #1
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Green's Theorem Problem

Need some steps and the solution

Use Green's theorem to show that the area of a region R bounded by the
closed curve C is given by

Check the attachment
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October 6th, 2017, 01:37 AM   #2
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Green's theorem states

$\displaystyle \int \int_R \left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right) = \int_C~P~dx + Q~dy$

let $P=0,~Q=x$

$\dfrac{\partial Q}{\partial x} =1,~\dfrac{\partial P}{\partial y}=0$

$\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}=1$

$\displaystyle \int\int_R ~1~ dA = \int\int_R ~\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}~dA = \int_C (0)dx + (x)dy = \int_C x ~dy$

I leave you to find $P,~Q$ that allows you to get the other form they ask for
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