GATE Electrical Engineering (EE) 2018 Solved Paper

Question : 57 of 65

Marks: +1, -0

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

∫↙{c}(y^{2}+2 y x) d x+(2 x y+x^{2}) d y

is__________(upto 2 decimal places).

Solution:

Consider a closed curve C and the coordinate axes.
Applying Green’s Theorem

∫↙{c}(y^{2}+2 y x) d x+(2 x y+x^{2}) d y=∫∫(\;{∂ N}/{∂ x}-\;{∂ M}/{∂ y}) d x d y

\;{∂ N}/{∂ x}=\;{∂(2 x y+x^{2})}/{∂ x}=2 x+2 y

\;{∂ M}/{∂ y}=\;{∂(y^{2}+2 x y)}/{∂ y}=2 y+2 x

\;{∂ N}/{∂ x}-\;{∂ M}/{∂ y}=0

So,

∫↙{C}(y^{2}+2 y x) d x+(2 x y+x^{2}) d y=0

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