IGNOU PHE 4 2018 - Using Green’s Theorem evaluate the integral ∫C ( y2dx + 3xyd

Question

Using Green’s Theorem evaluate the integral ∫C ( y²dx + 3xydy)

where C is a semi

circle of radius 1 in the upper half plane, centered at the origin.

25 Jan 2018

Answer :

Word Count : 330

Using Green’s Theorem, we can evaluate the given line integral by converting it into a double integral over the enclosed region. ### Step 1: Green’s Theorem Green’s Theorem states that for a simple, closed curve \( C \) enclosing a region \( R \): \[ \oint_C \left( P dx + Q dy \right) = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \] where: - \( P = y^2 \) - \( Q = 3xy \) ### Step 2: Compute Partial Derivatives \[ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}

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