Answer
$ \int_{a}^b f(x) \ dx =- \oint_{C} y \ dx $ (proved)
Work Step by Step
The tangential form for Green Theorem is given as:
Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy$
Since, $\int_{a}^b f(x) \ dx =\int_{a}^b \int_{0}^f(x) \ dy \ dx =\iint_{R} \ dx \ dy$
Now, $Area = \iint_{R} \ dx \ dy = \iint_{R} 0+1 \ dx \ dy= \iint_{R} (\dfrac{\partial (0)}{\partial x} +\dfrac{\partial (y)}{\partial y}) dx dy $
or, $ = \oint_{C} 0 \ dy - y \ dx $
or, $ \int_{a}^b f(x) \ dx =- \oint_{C} y \ dx $ (proved)
