Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ (this is not the general case, let me keep the notation simple).
In this way we are "freezing" the index $n$, and the integral is able to compute $c_n$.
This is due to the fact that $\int_{-\pi}^{\pi}e^{ikx}dx=0$ for all $k\ne0$. I am able to solve this integral, but I would like to "see" it, as a kind of geometric proof.
So my question is: which is the geometrical meaning of $\int_{-\pi}^{\pi}e^{ikx}dx$?
Should I draw a three dimensional curve and then calculate the area of a surface? It is not obvious to me that this area should be equal to zero.