Visual U-Substitution

Last week, I wrote about u-substitution and ended by wondering what it would look like to give students a visual representation of what it does to the graph. I came up with this.

The red function is f(x) = \frac{4x}{\sqrt{2x^2-1}}, and I’ve shaded in the area represented by \int_2^3 f(x) dx .

The blue function is g(x) = \frac{1}{\sqrt{x}}, and I’ve shaded in the area represented by \int_7^{17} g(x) dx .

When you work through u-substitution on the first integral and change your bounds, you come up with the second integral and the new bounds. If you estimate the shaded area, you’ll see that they’re both just under 12 blocks.

When we do u-substitution, we’re not just doing some algebraic trick. (Well, yes, ok, it is an algebraic trick, but it’s not just an algebraic trick!) We’re doing a transformation that gives us a new function that’s easier to integrate (when we change x to u), then undoing the transformation by changing it back into the old one (when we change u back to x). If we’re doing a definite integral, then we’re finding a new function that’s easier to integrate, but doing the new definite integral over some bounds that give us the same area.

I don’t necessarily expect my students to walk away with a deep understanding of transformation of the x-axis from this. I’m still working out the details of it myself – most of the transformations I’ve learned about (other than the ubiquitous function transformations) were of the complex plane, and that was a while ago and fuzzy to begin with. But I like that this reinforces that we’re finding area, and that we’re changing the integral in such a way that the area doesn’t change.

Also, my students often struggle with why we have to change the bounds. I think this visual might help instead of me babbling about transformations of the x-axis and finding our new u-coordinates.

I didn’t think of this until after school on of our last day of u-substitution, so I showed it to my students the next day, after going over their homework. That was fine. I think it may have helped someone solidify their understanding. At the very least, it’ll give them something to chew on. I didn’t really have any questions for them, though – I did all the talking. And we weren’t really in the moment when they were wondering about it any more.

But next year, I want to show it to them when we start u-substitution with definite integrals. And I’ll start by showing them the red graph and the shaded area. We’ll work through the u-substitution together and come up with the blue graph, then I’ll have them find the bounds that give the same area – maybe with sliders. I need some functions that give friendlier areas, instead of “just under 12 blocks.” Then ask how the new bounds relate to the old bounds, and they can find the connection. This way, they’re doing more thinking, and we start with the idea of new bounds but same area, instead of coming back to it at the end.

Clearly, this is turning into a Desmos Activity.

If I ever build it, I’ll post a link here.



4 thoughts on “Visual U-Substitution

  1. A while back, Sam Shah shared a nice set of applets he made in GeoGebra:

    U-Substitution (and chain rule) has been on my list of things I want to build a visual for for a while now; I really like your approach, but there’s got to be a good way to /show/ that the two areas really are the same.
    Hmmm you got me thinking about non-digital representations of this… I want to pour sand into one curve, then open a valve and have it flow into the other curve. But that’d only be possible if you already know the u function, and I’d rather have the visual *generate* that function… Ahh such a good problem!


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