Answer
Counterclockwise Circulation =$\dfrac{-44}{15}$
Work Step by Step
The tangential form for Green's Theorem - Counterclockwise Circulation can be defined as: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy $
So, $M=(y+e^x \ln y) ; N=\dfrac{e^x}{y}$
Now, $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial (\dfrac{e^x}{y})}{\partial x}-\dfrac{\partial (y+e^x \ln y)}{\partial y}) dx dy$
or, $\iint_{R} (\dfrac{e^x}{y}-1-\dfrac{e^x}{y}) dx dy =- \iint \space dx \space dy$
or, $=-\int_{-1}^{1} \int_{x^4+1}^{3-x^2} dy dx$
or, $=-\int_{-1}^{1} [y]_{x^4+1}^{3-x^2}$
or, $=-\int_{-1}^{1} (2-x^2-x^4) dx$
or, $=\dfrac{-44}{15}$
