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GATE 2019 EC – Question 37

Engineering Mathematics · Vector Analysis · 2 marks · Multiple choice

Consider the line integral $\oint_C(x\,dy-y\,dx)$, the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $R$ shown in the figure below. The region $R$ is the area enclosed by the union of a $2\times3$ rectangle and a semi-circle of radius 1. The line integral evaluates to

Diagram for GATE 2019 EC question 37
  1. $6+\pi/2$
  2. $8+\pi$
  3. $12+\pi$
  4. $16+2\pi$

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Show answer and explanation

Correct answer: (C) $12+\pi$

Explanation

By Green's theorem, $\oint(x\,dy-y\,dx)=2\times\text{area}$. The area is $2\times3+\frac\pi2=6+\frac\pi2$, so the integral is $12+\pi$.