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Length of Subnormal

Suppose C is ...

Question

Suppose C is the closed curve defined as the circle $x^{2} + y^{2} = 1$ with C oriented anti clockwise. The value of $\oint (x y^{2} d x + x^{2} y d y)$ over the curve C equals ______
0

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Solution

The correct option is A 0
Using Green's theorem,
$\oint M d x + N d y = \iint_{R} (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) d x d y$
$\int (x y^{2} d x + x^{2} y d y) = \int \int (2 x y - 2 x y) d x d y = 0$

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