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By Green's theorem, the area of a region \( R \) equals

  • A.\( \displaystyle\oint_{C}\left(x\,dy + y\,dx\right) \)
  • B.\( \tfrac{1}{2}\displaystyle\oint_{C}\left(x\,dx + y\,dy\right) \)
  • C.\( \displaystyle\oint_{C}\left(x\,dx - y\,dy\right) \)
  • D.\( \tfrac{1}{2}\displaystyle\oint_{C}\left(x\,dy - y\,dx\right) \)

Answer: D

Take \( M = -y \) and \( N = x \), so that \( \partial N/\partial x - \partial M/\partial y = 1 + 1 = 2 \):

\[ \oint_{C}\left(x\,dy - y\,dx\right) \;=\; \iint_{R} 2\,dA \;=\; 2A. \]

Hence the half. This is how a planimeter measures an area from its boundary alone, without ever sampling the interior

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