Chapter 14.4, Problem 58E

Green’s Theorem as a Fundamental Theorem of Calculus

Show that if the circulation form of Green’s Theorem is applied to the vector field $\langle 0,\frac{f\left(x\right)}{c}\rangle$ and $R=\left\{\left(x,y\right):a\le x\le b,0\le y\le c\right\}$, then the result is the Fundamental Theorem of Calculus,

${\displaystyle {\int }_a^b\frac{df}{dx}dx}=f\left(b\right)-f\left(a\right).$