In Complex Analysis texts, one often sees integrals of functions $f : A \subset \mathbb{C} \to \mathbb{C}$ along contours $\gamma : [0,1] \to \mathbb{C}$.
But I've never seen discussion of integrals of functions $f : A \subset \mathbb{C} \to \mathbb{C}$ over domains $D \subset A$ in any of these texts.
It seems like it should be possible to define such integrals, so why aren't they discussed in most complex analysis textbooks? Are there any major results about them like, for example, Cauchy's integral formula for contour integrals? Are there any textbooks that do delve into these kinds of integrals?
Examples
By a contour integral, I mean something like the following: consider the contour $\gamma : [0,1] \to \mathbb{C}$ given by $\gamma(t) = e^{2\pi i t}$ and the function $f : \mathbb{C} – \{0\} \to \mathbb{C}$ given by $f(z) = \frac{1}{z}$. The integral of $f$ around the contour $\gamma$ is
$$
\int_\gamma f = \int_0^1 f(\gamma(t))\gamma'(t) dt = \int_0^1 \frac{1}{e^{2\pi i t}} \frac{2\pi i e^{2\pi i t}}{1} dt = \int_0^1 2\pi i dt = 2\pi i
$$
By an integral over a domain, I mean something like the following: consider the function $g : \mathbb{C} \to \mathbb{C}$ given by $g(z) = z^2$ and the domain $D = \{z \in \mathbb{C} : |z| \leq 1\}$. The integral of $g$ over the domain $D$ is
$$
\int_D g d\mu \stackrel{?}{=}
$$
where, presumably, $d\mu$ is something like the Haar measure on $\mathbb{C}$ with $\mu([0,1]\times[0,1]) = 1$.
Best Answer
The reason contour integrals are useful in complex analysis is due to Cauchy theorem and all its corollaries like the computation of integrals with residues, the Cauchy bounds for derivatives of all orders, maximum modulus theorem, very strong results about locally uniform convergence of holomorphic functions (such can be differentiated term by term for example which is not the case for uniform convergence of even real analytic functions) etc.
2-dimensional integrals have their use here and there in Complex Analysis (see Area Theorem for univalent functions which leads to the original Bieberbach estimate of the second coefficient) and various other specialized topics like the Bergman kernel and various spaces related to the Hardy spaces but defined by area integrals bounds, however, they are considerably more important in Real Analysis topics (in 2 variables here)