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GATE 2022 EE – Question 51

Engineering Mathematics · Calculus: Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes's theorem, Gauss's theorem, Divergence theorem, Green's theorem · 2 marks · Multiple choice

Let $R$ be a region in the first quadrant of the $xy$ plane enclosed by a closed curve $C$ considered in counter-clockwise direction. Which one of the following expressions does not represent the area of the region $R$?

Diagram for GATE 2022 EE question 51
  1. $\displaystyle\iint_Rdx\,dy$
  2. $\displaystyle\oint_Cx\,dy$
  3. $\displaystyle\oint_Cy\,dx$
  4. $\displaystyle\frac12\oint_C(x\,dy-y\,dx)$

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Correct answer: (C) $\displaystyle\oint_Cy\,dx$

Explanation

By Green's theorem, $\oint_Cx\,dy=\iint_R1\,dA=A$ and $\oint_Cy\,dx=-\iint_R1\,dA=-A$ for a counter-clockwise curve. So (B) and (D) give the area, (A) is the area by definition, and (C) gives $-A$, which does not represent the area.