# Alternative ways of thinking about the one-variable Riemann integral for elementary calculus,

I think I've done a decent job with teaching my students limits and derivatives so far in elementary calculus -- they were particularly intrigued with how easy and how accurate a first-order, linear approximation can be.

We'll soon start on integration, and I am wondering if I could give them an alternative way of thinking about the Riemann integral other than that "it is the area under the curve". What way of thinking about the Riemann integral in one variable would surprise them a bit?

(They are mostly freshmen and have seen calculus in some form in their high school days, it seems, but I don't think any of them would have any numerical methods background for me to discuss numerical integration with them.)

• Probability with a continuum of outcomes. E,g, Buffon's Needle Problem. – DanielWainfleet Oct 18 '17 at 7:53
• @DanielWainfleet Nice idea. But Buffon is better done calculus free: cs.umb.edu/~eb/piday/whypi.pdf – Ethan Bolker Oct 18 '17 at 12:42

Perhaps start with Riemann sums to find distance covered when velocity is known but you can't guess an antiderivative - something they might appreciate if they are fond of derivatives. Gives you a head start on the fundamental theorem of calculus.

Then you can point out that the same formalism finds areas.

You can finesse the numerical methods part by demonstrating a simple (easy to read Python) program or spreadsheet that finds Riemann sums with the number of division points as an input parameter.

• Hi @EthanBolker, my students really liked a recent quiz question involving velocities and accelerations - I think they would really appreciate this suggestion of yours. Thanks so much! – D.Hutchinson Oct 18 '17 at 2:46

I find people are amazed, when they see it, that Euler's method works.

In a way it is intuitive in a similar manner to how first order approximation works, and makes it less focused on abstract limits and convergence.

And, there is a hit movie that centers on it (Hidden Figures)

• Hi @eSurfsnake, you're suggesting that I show them that integration solves a simple differential equation, retrieving a position function, so that we can analyze where the particle is / what its trajectory looks like? If so, that sounds pretty awesome, compared to the mundane "area under the curve" concept ... – D.Hutchinson Oct 18 '17 at 2:45
• That is it. It always feels like the mathematical equivalent of building a long bridge by making the first foot cantilever out, then adding another foot to that, until you reach the other side. It's a lot more interesting than area under the curve, which itself is sort of a dated fascination with centroids and the like. – eSurfsnake Oct 18 '17 at 3:01