We saw in the last video that the area under a curve is a limit of
a sum. In general, such a limit is called a definite integral.
Here is the formal definition.

If $f$ is a function defined on $a \leq x \leq
b$, we divide the interval $[a,b]$ into $n$ subintervals of equal
width: $\Delta x = \frac{b-a}{n}$. Let ${x_i}^*$ be any sample
point in the $i$th interval. Then the definite integral of $f$
from $a$ to $b$ is
$$\int_a^b f(x)\, dx = \lim_{n \rightarrow \infty}\, \sum_{i=1}^n\, f({x_i}^*)\,
\Delta x$$
provided that limit exists and gives the same value for all
possible choices for ${x_i}^*$. If it does exist, we say that $f$ is
integrable on $[a,b]$.

If
$f(x) \ge 0$ for all $x$ in $[a,b]$, then $\int_a^b f(x)\, dx$ represents the
area under the curve $y=f(x)$ between $x=a$ and $x=b$. But integrals are not just areas, and make sense even when $f(x)$ isn't positive.