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 the above function, \int_{c} F \cdot d l along the curve C, which follows the parabola y=x^{2} as shown below is ________(rounded off to 2 decimal places).

Correct Answer is : -3.05- -2.95

GATE Electrical Engineering (EE) 2020 Solved Paper

Question : 37 of 65

Marks: +1, -0

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 the above function,

∫↙{c} F ⋅ d l

along the curve

C

, which follows the parabola

y=x^{2}

as shown below is ________(rounded off to 2 decimal places).

Solution:

f=y a_{x}-x a_{y}

∫↙{c} f . d ℓ=∫↙{c}(y a_{x}-x a_{y}) ⋅(d x a_{x} ± d y a_{y})

=∫↙{c}(y d x-x d y)

But

y=x^{2}

d y=2 x d x

∫↙{c} f . d ℓ=∫↙{c} x^{2} d x-x(2 x d x)

=∫ x^{2} d x-2 x^{2} d x=-∫ x^{2} d x

=-∫↙{-1}↖{2} x^{2} d x=-[\;{x^{3}}/{3}]_{-1}^{2}

=-[\;{8}/{3}-(-\;{1}/{3})]=-3

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