Answer
(a) $0$
b) $(h-k) \times ( \ Area \ enclosed \ inside \ the \ curve \ C)$
Work Step by Step
The tangential form for Green Theorem is given as:
Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) \ dx \ dy ....(1)$
(a) $\oint_C f(x) dx +g(y) dy= \iint_{R} (0-0) dx dy $
or, $\oint_C f(x) dx +g(y) dy=0$
(b) $\oint_C k(y) \ dx +h(x) \ dy= \iint_{R} (\dfrac{h(x) }{\partial x}-\dfrac{k (y)}{\partial y}) \ dx \ dy \\=(h-k) \iint_{R} \ dx \ dy \\=(h-k) \times ( \ Area \ enclosed \ inside \ the \ curve \ C)$
