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GATE 2023 EE – Question 62

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 · Numerical answer

The closed curve shown in the figure is described by $r=1+\cos\theta$, where $r=\sqrt{x^2+y^2}$; $x=r\cos\theta$, $y=r\sin\theta$. The magnitude of the line integral of the vector field $F=-y\hat i+x\hat j$ around the closed curve is _____________ (Round off to 2 decimal places).

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Correct answer: 9.37 to 9.47

Explanation

By Stokes' theorem, $\oint F\cdot dl=\iint(\nabla\times F)\cdot dA=2\times\text{Area}$, since $\nabla\times F=2\hat k$. The area of the cardioid $r=1+\cos\theta$ is $\dfrac12\int_0^{2\pi}(1+\cos\theta)^2d\theta=\dfrac{3\pi}2$. So the integral is $3\pi=9.42$.