Let a_x and a_y be unit vectors along x and y directions, respectively. A vector function is given by F = a_x y - a_y x The line integral of above function

${\int }_{C}F\cdot dl$ Along the curve C, which follows the parabola y = x2 as shown below is ______ (rounded off to 2 decimal places)

Answer :

Ans. – 3

Given that,

F = a_x y - a_y x

y = x²

By differentiating on both sides,

⇒ dy = 2x dx

$\int F.dl={\int }_C\left(a_{x}y-a_{y}x\right).\left(dx{a}_x+dy{a}_y\right)$

$={\int }_C\left(ydx-xdy\right)$

In the given graph, the limits of x are: -1 to 2.

$\int F.dl={\int }_{-1}^2{x}^{2}dx-2x^{2}dx$

$=-{\int }_{-1}^2{x}^{2}dx=-\frac{1}{3}{\left[x^3\right]}_{-1}^2=-3$