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GATE 2018 EE – Question 52

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

As shown in the figure, $C$ is the arc from the point $(3,0)$ to the point $(0,3)$ on the circle $x^2+y^2=9$. The value of the integral $\int_C(y^2+2yx)\,dx+(2xy+x^2)\,dy$ is ________ (up to 2 decimal places).

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Correct answer: 0

Explanation

The field is conservative because $\frac{\partial}{\partial y}(y^2+2xy)=2y+2x=\frac{\partial}{\partial x}(2xy+x^2)$. A potential is $\phi=xy^2+x^2y$, and $\phi(0,3)-\phi(3,0)=0-0=0$.