14.1 Integration in theory

Let’s build a solid theoretical foundation for the intuitive explanation! Let f : [a,b] →ℝ be an arbitrary bounded function, and our goal is to calculate the signed area under the graph. (Note that the signed area is negative if the graph goes below the x axis. In the time-speed graph example above, this is equivalent to moving backward, thus decreasing the distance traveled from the starting point.)

Let a = x₀/span>x₁/span>…/span>x_n = b an arbitrary partition of the interval [a,b].

For notational convenience, we’ll denote this partition as X = {x₀,…,x_n} as well. The granularity (or mesh) of X is defined by

which is the length of the biggest gap in X. Note that the partition is not necessarily uniform, so jx_i −x_i−1j is not constant.

We are going to use an argument similar to the squeeze principle (Corollary 3) to make the approximation idea rigorous. (You know, the one where we replaced...