The definition of area under curve

Let $f$ be a continuous function on interval $[a,b]$. Divide $[a,b]$ into $n$ subintervals of length $\Delta x = (b-a)/n$. Choose (sample) points $x_i^*$ in $i$th interval, for each $i$. The (signed) area between the graph of $f$ and the $x$ axis is approximately

$$A_n \sim f(x_1^*) \Delta x + \cdots + f(x_n^*) \Delta x$$

$$= \sum_{i=1}^n f(x_i^*) \Delta x.$$

(The $\sum$ is notation to make it easier to write down and think about the sum.)

Definition 2.1.1 (Signed Area)   The (signed) area between the graph of $f$ and the $x$ axis between $a$ and $b$ is

$$\lim_{n\to\infty} \left( \sum_{i=1}^n f(x_i^*) \Delta x \right)$$

(Note that $\Delta x = (b-a)/n$ depends on $n$.)

It is a theorem that the area exists and doesn't depend on the choice of $x_i^*$.