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GATE 2020 EE – Question 54

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

Let $\mathbf a_x$ and $\mathbf a_y$ be unit vectors along x and y directions, respectively. A vector function is given by

$$\mathbf F=\mathbf a_xy-\mathbf a_yx$$

The line integral of the above function $\int_C\mathbf F\cdot d\mathbf l$ along the curve $C$, which follows the parabola $y=x^2$ as shown below is ________ (rounded off to 2 decimal places).

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Correct answer: -3.05 to -2.95

Explanation

On the parabola $y=x^2$, $dy=2x\,dx$, so $\mathbf F\cdot d\mathbf l=y\,dx-x\,dy=x^2dx-2x^2dx=-x^2dx$. Integrating from $x=-1$ to $x=2$ gives $-\frac{x^3}{3}\Big|_{-1}^{2}=-\frac{8+1}{3}=-3$.