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GATE 2024 IN – Question 46

Engineering Mathematics · Calculus and vector calculus · 2 marks · Multiple choice

Let $C$ be the counterclockwise boundary of the rectangle with vertices $(0,0),(2,0),(2,1),(0,1)$. The value of $\oint_C(-e^y\,dx+e^x\,dy)$ is

  1. $e^2+2e-3$
  2. $e^2-2e-3$
  3. $e^2+e-1$
  4. $e^2+e+1$

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Correct answer: (A) $e^2+2e-3$

Explanation

Green’s theorem gives $\oint_C P\,dx+Q\,dy=\int_0^2\int_0^1(\partial_xQ-\partial_yP)\,dy\,dx$. Here the integrand is $e^x+e^y$. The two contributions are $e^2-1$ and $2(e-1)$, totalling $e^2+2e-3$, **A**.