In real analysis, we can understand integration as areas under the curve. In complex analysis, I saw many theorems (Cauchy's theorem, Goursat's theorem etc.) about complex integration, but I couldn't see its relation with "area".
Q. What should one basically understand by the complex integration, and what should we think when there is some discussion of complex integration in a book.
They are not related with the area but are related with some other intuitive things. Write $f=v_1-iv_2$, $dz=dx+idy$, and think of $(v_1,v_2)$ as components of velocity of a flow of an ideal fluid. Then $$\int f(z)dz=\int v_1dx+v_2dy+i\int v_1dy-v_2dx=:A+iB.$$ If the curve of integration is a simple closed curve then $A$ is the circulation along $\gamma$ and $B$ is the flux through $\gamma$ of your fluid flow.
