Summary
We have seen that given a positive valued function $f$ on an
interval $[a,b]$, the area under $f$ 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. (We will use the right
endpoint $x_i$ for the $x_i^*$ in our work.)
We have seen that the larger the value of $n$, the better the
approximations. We are now 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 definite
integral of $f(x)$ from $a$ to $b$, and is denoted
$$\int_a^b f(x)\, dx.$$ We will work with the definite integral in
the next module.
