Answer
$6 \pi$
Work Step by Step
Here, we have $f(x, y)=2x -3y$ and $g (x, y)=3x+4y$
This gives: $\dfrac{\partial f}{\partial x}=2$ and $\dfrac{\partial g}{\partial y}=4$
Next, we will use the Green's Theorem to compute the vector field.
$\oint_C (f dy-g dx)=\int_R (\dfrac{\partial f}{\partial x}+\dfrac{\partial g}{\partial y}) \ dA$
or, $=\int_{R} 6 \ dA$
Here, $R$ defines a circular region.
So, we can write as: $\oint_C (f dy-g dx)=\int_{R} 6 \ dA= (6) \times (1)^2 \times \pi=6 \pi$
