The integral -cydx-xdy is evaluated along the circle x2-y2-1-4 traversed in counter clockwise direction- The integral is equal to

Applying Green's theorem ∮_c ( F_1dx+F_2dy )= ∬_R ( ∂ F_2/∂ x ∂ F_1/∂ y )dxdy ∮_cydx xdy=∬_R( 1 1)dxdy = 2∬_Rdxdy = 2× (area of circle) = 2×π ( 1/2 )^2 ∮_cyd ...

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Byju's Answer

Standard XII

Mathematics

Obtaining Centre and Radius of a Circle from General Equation of a Circle

The integral ...

Question

The integral $\oint_{c} (y d x - x d y)$ is evaluated along the circle $x^{2} + y^{2} = \frac{1}{4}$ traversed in counter clockwise direction. The integral is equal to

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\frac{π}{4}

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\frac{π}{2}

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\frac{π}{4}

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Solution

The correct option is C - $\frac{π}{2}$
Applying Green's theorem
$\oint_{c} (F_{1} d x + F_{2} d y) = \iint_{R} (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d x d y$
$\oint_{c} y d x - x d y = \iint_{R} (- 1 - 1) d x d y$
$= - 2 \iint_{R} d x d y$
$= - 2 \times (a r e a o f c i r c l e)$
$= - 2 \times π {(\frac{1}{2})}^{2}$
$\oint_{c} y d x - x d y = \frac{- π}{2}$

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The integral ∮c(ydx−xdy) is evaluated along the circle x2+y2=14 traversed in counter clockwise direction. The integral is equal to

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The integral $\oint_{c} (y d x - x d y)$ is evaluated along the circle $x^{2} + y^{2} = \frac{1}{4}$ traversed in counter clockwise direction. The integral is equal to