Summary
We have seen that given a positive valued function on an interval
$[a,b]$, the area under the curve on that interval can be approximated
by:
$$A \approx \sum_{i=1}^n f({x_i}^*)\, \Delta x$$
where ${x_i}^*$ is any $x$ value in the $i$th interval and $\Delta x$
is the length of each rectangle.
And we have seen that the larger the value of $n$, the better the
approximations.
We are ready to define the area under this curve:
$$A = \lim_{n \rightarrow \infty} \,\sum_{i=1}^n \,f(x_i) \,\Delta x$$
This limit is so important that we give it a special name and notation.
It is the integral of $f(x)$ from $a$ to $b$, and is denoted
$$\int_a^b f(x)\, dx.$$
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