We saw previously 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 $[x_{i-1},x_i]$ of equal width $\Delta x = \frac{b-a}{n}$. 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. 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 make sense even when $f(x)$ isn't positive, as you will see in the video below.
Warning:
The definite integral of a
function $f$, sometimes just called the integral
of $f$, is not equal to an
antiderivative of $f$. An antiderivative of
$f$ is a function $F$ such that
$F'=f$. The definite integral of $f$ is a number, which can be viewed as
representing the (positive and negative) area between $f$ and
the $x$-axis.
However, these two different concepts, the
definite integral and the antiderivative, are related by the
beautiful Fundamental Theorem of
Calculus, which we will see in the next module.