Other then as a fantastic tool to evaluate some difficult real integrals, complex integrals have many purposes.
Firstly, contour integrals are used in Laurent Series, generalizing real power series.
The argument principle can tell us the difference between the poles and roots of a function in the closed contour $C$:
$$\oint_{C} {f'(z) \over f(z)}\, dz=2\pi i (\text{Number of Roots}-\text{Number of Poles})$$
and this has been used to prove many important theorems, especially relating to the zeros of the Riemann zeta function.
Noting that the residue of $\pi \cot (\pi z)f(z)$ is $f(z)$ at all the integers. Using a square contour offset by the integers by $\frac{1}{2}$, we note the contour disappears as it gets large, and thus
$$\sum_{n=-\infty}^\infty f(n) = -\pi \sum \operatorname{Res}\, \cot (\pi z)f(z)$$
where the residues are at poles of $f$.
While I have only mentioned a few, basic uses, many, many others exist.
From context, it appears that $R(\gamma)$ is simply the range of the function $\gamma: [a,b] \to \mathbb{C}$. In other words, it's all the points that lie on the curve itself, not the interior or the closure of the interior.
Best Answer
$$\int_\gamma \bar{z}dz = \int_\gamma (x - iy)(dx + idy) = \int_\gamma (xdx + ydy) + i \int_\gamma( xdy - ydx)$$now hit this with stokes:$$ = \int_D d(xdx + ydy) + i\int_D d(xdy - ydx) = i \int_D 2 dx \wedge dy$$