How can some integrals be zero?

Question

This could be a nonsense but I have to try.

We are used to know that an integration like $$\int_a^b F(x)\ \text{d}x$$

gives us the area under the curve $F(x)$ from $a$ to $b$. The question is then: how is it possible for an area to be zero?

Answer 1

Because an integral is not just "area," but signed area. That is to say, if $F(x) > 0$ on some interval $[c,d]$, then the integral $$\int_{x=c}^d F(x) \, dx > 0.$$ If $F(x) < 0$ on another interval $[c', d']$, then $$\int_{x=c'}^{d'} F(x) \, dx < 0.$$ Therefore, for a function that takes on positive and negative values in some interval $[a,b]$, it is entirely possible for the integral of $F$ on $[a,b]$ to be equal to $0$.

Answer 2

$$\int_0^{2\pi} sin(x) dx = 0$$