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.