I see for the other A To Z I did this year I did something else named for Riemann. So I did. Bernhard Riemann did a lot of work that’s essential to how we see mathematics today. We name all kinds of things for him, and correctly so. Here’s one of his many essential bits of work.
The Riemann Sum is a thing we learn in Intro to Calculus. It’s essential in getting us to definite integrals. We’re introduced to it in functions of a single variable. The functions have a domain that’s an interval of real numbers and a range that’s somewhere in the real numbers. The Riemann Sum — and from it, the integral — is a real number.
We get this number by following a couple steps. The first is we chop the interval up into a bunch of smaller intervals. That chopping-up we call a “partition” because it’s another of those times mathematicians use a word the way people might use the same word. From each one of those chopped-up pieces we pick a representative point. Now with each piece evaluate what the function is for that representative point. Multiply that by the width of the partition it was in. Then take those products for each of those pieces and add them all together. If you’ve done it right you’ve got a number.
You need a couple pieces in place to have “the” Riemann Sum for something. You need a function, which is fair enough. And you need a partitioning of the interval. And you need some representative point for each of the partitions. Change any of them — function, partition, or point — and you may change the sum you get. You expect that for changing the function. Changing the partition? That’s less obvious. But draw some wiggly curvy function on a sheet of paper. Draw a couple of partitions of the horizontal axis. (You’ll probably want to use different colors for different partitions.) That should coax you into it. And you’d probably take it on my word that different representative points give you different sums.
Very different? It’s possible. There’s nothing stopping it from happening. But if the results aren’t very different then we might just have an integrable function. That’s a function that gives us the same Riemann Sum no matter how we pick representative points, as long as we pick partitions that get finer and finer enough. We measure how fine a partition is by how big the widest chopped-up piece is. To be integrable the Riemann Sum for a function has to get to the same number whenever the partition’s size gets small enough and however we pick points inside. We get the lovely quiet paradox in which we add together infinitely many things, each of them infinitesimally tiny, and get a regular old number out of all that work.
We use the Riemann Sum for what we call numerical quadrature. That’s working out integrals on the computer. Or calculator. Or by hand. When we do it by evaluating numbers instead of using analysis. It’s very easy to program. And we can do some tricks based on the Riemann Sum to make the numerical estimate a closer match to the actual integral.
And we use the Riemann Sum to learn how the Riemann Integral works. It’s a blessedly straightforward thing. It appeals to intuition well. It lets us draw all sorts of curves with rectangular boxes overlaying them. It’s so easy to work out the area of a rectangular box. We can imagine adding up these areas without being confused.
We don’t use the Riemann Sum to actually do integrals, though. Numerical approximations to an integral, yes. For the actual integral it’s too hard to use. What makes it hard is you need to evaluate this for every possible partition and every possible pick of representative points. In grad school my analysis professor worked through — once — using this to integrate the number 1. This is the easiest possible thing to integrate and it was barely manageable. He gave a good try at integrating the function ‘f(x) = x’ but admitted he couldn’t do it. None of us could.
When you see the Riemann Sum in an Introduction to Calculus course you see it in simplified form. You get partitions that are very easy to work with. Like, you break the interval up into some number of equally-sized chunks. You get representative points that follow one of a couple good choices. The left end of the partition. The right end of the partition. The middle of the partition.
That’s fine, numerically. If the function is integrable it doesn’t matter what partition or representative points we pick. And it’s fine for learning about whether functions are integrable. If it matters whether you pick left or middle or right ends of the partition then the function isn’t integrable. The instructor can give functions that break integrability based on a given partition or endpoint choice or whatever.
But that isn’t every possible partition and every possible pick of representative points. I suppose it’s possible to work all that out for a couple of really, really simple functions. But it’s so much work. We’re better off using the Riemann Sum to get to formulas about integrals that don’t depend on actually using the Riemann Sum.
So that is the curious position the Riemann Sum has. It is a fundament of integral calculus. It is the way we first define the definite integral. We rely on it to learn what definite integrals are like. We use it all the time numerically. We never use it analytically. It’s too hard. I hope you appreciate the strange beauty of that.