Question

Evaluate the following integral over the ellipse ​

Answer 1

The underlying vector field,

F(x, y) = -y/(4x ² + 9y ²) i + x/(4x ² + 9y ²) j,

is conservative, so any integral of F over a closed path is 0.

To establish that F is conservative, we want to find a scalar function f(x, y) whose gradient is equal to F(x, y), which entails solving

$\dfrac{\partial f}{\partial x}=-\dfrac y{4x^2+9y^2}$

$\dfrac{\partial f}{\partial y}=\dfrac x{4x^2+9y^2}$

Integrating the first equation with respect to x yields

$f(x,y)=-\dfrac16\arctan\left(\dfrac{2x}{3y}\right)+g(y)$

and differentiating with respect to y gives

$\dfrac x{4x^2+9y^2}=\dfrac x{4x^2+9y^2}+\dfrac{\mathrm dg}{\mathrm dy} \implies \dfrac{\mathrm dg}{\mathrm dy}=0 \implies g(y)=C$