Back in Winter Storm Warning, we estimated that the total accumulation (net change) of snow over the $4$ hours was the sum of a bunch of terms that all looked like $f'(t) \Delta t$, as these were the amount of snow that fell over a short time period if we assumed a constant rate (and computed it as $(\text {rate}) \times (\text {change of time})$). Adding these up gave us an approximation of the overall change over the $4$ hours. The notation for this summation is $\sum _i f'(t_i)\Delta t$, where the “$i$’s” just mean that we take a sequence of times over the same size interval (i.e., the $t_i$’s were the left-hand endpoints of each interval. We will be not concerning ourselves with this “$i$” notation anymore here- just remember that what we are plugging a sequence of $t$’s into while the change in $t$ (length of each interval) remains the same).

We also saw that if the length of the intervals ($\Delta t$) gets very small, our approximation gets closer and closer to the exact value (why?), which we wrote as $\int _a^b f'(t) \, dt$, where $[a, b]$ is the interval of time we were interested in. Thus, $\sum f'(t) \Delta t$ “turns into” $\int _a^b f'(t)\, dt$ as $\Delta t$ gets very small. And they were also approximating (and eventually exactamating) the area between the graph of $f'(t)$ and the $t$-axis.

Well, it turns out that if we replace $f'(t)$ by something else (with the goal of doing a different sort of measurement) and the measurement involves adding up the products of (“that something else”) $\times \Delta t$ and the estimation becomes more and more exact as $\Delta t$ gets closer to zero, then it turns out that the sum $\sum (\text {something else})$ “turns into” $\int _a^b (\text {something else})\, dt$. These types of sums are called “Riemann Sums”, named after a famous $19$th century mathematician named Bernhard Riemann. The biggest problem in most of these applications is defining the “something”.

The following list of activities contain just a couple of a few applications of Riemann sums in which we use integrals to do more than just measure area: See the activities Stuck in the Middle , It Slices and It Dices , SoSo , and It’s a Long Way to Tipperary! .