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Thread: Green's theorem

  1. #1
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    Green's theorem

    Hey, I have a problem with solving this:

    With Green's theorem solve this problem:
    Green's theorem-screenshot_1.png

    Can anybody help me, please?
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  2. #2
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    Re: Green's theorem

    consider the curve $C$ consisting of

    the line segment from (0,0) to $A$, the curved line from $A$ to $B$ and the segment from $B$ to $(0,0)$

    and the region $D$ enclosed by $C$ and apply Green's Theorem
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  3. #3
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    Re: Green's theorem

    I tried this, and I don't know how to continue...
    Green's theorem-screenshot_1.png
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  4. #4
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    Re: Green's theorem

    The path $L_1,L_2,L_3,L_4$ leads to the integral

    $$\int _0^{\sqrt{3}}\int _0^{\sqrt{4-y^2}}3\left(x^2+y^2\right)dxdy$$

    which is a little difficult to evaluate

    on the other hand the path $A\longrightarrow B\longrightarrow O\longrightarrow A$

    will result in an integral which is easier to evaluate (using polar coordinates)

    $$\int _0^{\pi /3}\int _0^23r^3drd\theta $$



    Spoiler:
    Final answer: $$4 \pi -\frac{\sqrt{3}}{2}$$
    Thanks from damir99
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