GATE 2018 EE Q.42: solution and answer

GATE 2018 · EEQ.42 · 2 marks · NAT

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 $\int_{C} (y^{2} + 2 y x) d x + (2 x y + x^{2}) d y$ is ______ (up to 2 decimal places).

GATE 2018 EE question 42: The first quadrant of the x-y plane, with the x-axis pointing right and the y-axis pointing up. — Figure from the GATE 2018 EE paper. It is part of the question.

Answer and solution

ANSWER

The integral equals 0.00. The field is conservative with potential $ϕ = x y (x + y)$ , and $ϕ$ is 0 at both $(3, 0)$ and $(0, 3)$ .

Official answer: 0.0Source: Official answer key, GATE 2018 EE (IIT Guwahati)

Write the integral as $\int_{C} P d x + Q d y$ with $P = y^{2} + 2 x y$ and $Q = 2 x y + x^{2}$ .
Check whether the field is conservative. The integrand is exact when $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ . Here $\frac{\partial P}{\partial y} = 2 y + 2 x$ and $\frac{\partial Q}{\partial x} = 2 y + 2 x$ . They are equal at every point of the plane, and the plane is simply connected, so the integral does not depend on the path.
Find a potential $ϕ$ with $ϕ_{x} = P$ and $ϕ_{y} = Q$ . Integrating $P$ with respect to $x$ gives $ϕ = x y^{2} + x^{2} y + g (y)$ . Then $ϕ_{y} = 2 x y + x^{2} + g^{'} (y)$ , and setting this equal to $Q$ gives $g^{'} (y) = 0$ . So $ϕ (x, y) = x y^{2} + x^{2} y = x y (x + y)$
Apply the fundamental theorem for line integrals, $\int_{C} \nabla ϕ \cdot d r = ϕ (end) - ϕ (start)$ . The path starts at $(3, 0)$ and ends at $(0, 3)$ : $\int_{C} P d x + Q d y = ϕ (0, 3) - ϕ (3, 0) = 0 - 0 = 0$
Check: $ϕ = x y (x + y)$ is zero wherever $x = 0$ or $y = 0$ , so both endpoints give 0, and the direction of travel along the quarter circle makes no difference. The integral is $0.00$ .

KEY IDEA

If $\partial P / \partial y = \partial Q / \partial x$ on a simply connected region, then $P d x + Q d y$ is the exact differential of a potential $ϕ$ . The line integral is then $ϕ (end) - ϕ (start)$ , whatever the path. Here $ϕ = x y (x + y)$ is zero on both coordinate axes, and both endpoints lie on an axis.

WATCH OUT FOR

Parametrising the arc as $x = 3 cos t$ , $y = 3 sin t$ and grinding through the trigonometric integral, which wastes time and invites sign or limit errors. Another mistake is applying Green's theorem to the open arc as if it were a closed curve. The exactness test takes seconds and shows the answer is simply $ϕ (0, 3) - ϕ (3, 0) = 0$ .

From the GATE 2018 EE paper. Every question of this paper · Download the full paper · All 23 Vector calculus questions, with notes

GATE 2018 EE Q.42: solution and answer

Answer and solution

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