u-substitution, visually

I created some calculus Geogebra applet thingies last summer that I wanted to use last year. Alas, time ran short and we never got to use them. However since I’m no longer teaching calculus (at least not next year), I figured I’d throw them up in case anyone else out there finds them useful.

They deal with u-substitution. I’ve always had a problem with teaching it. Here’s how it goes… You have some integral in terms of x. You convert all the xs and dxs into us and dus. And viola! It works out. It’s very powerful. And it’s procedural. And kids have throughout the years learned this “substitution”-y thing works [1]. So kids tend to like it.

But here’s the thing. For my kids, it’s just a random method to evaluate an integral. They don’t conceptually understand what is going on… what this changing of variables is doing.

When I thought deeply about this, I realized what truly is happening is that we are transforming space… From the x-f(x) plane to a much convoluted u-f(u) plane. But it is through our particular choice of u that makes the change in space beautiful, because it turns something that looks particularly nasty and converts it into something that looks rather nice. Ish.

Here is a screenshot from one of my geogebra applets illustrating this (you can click on the screenshot to be taken to the applet):


We start with a pretty ugly function that we’re integrating. But by using this substitution to morph space, we end up with a much nicer function. I mean, throw both of these up and ask your kids — which one of these would they rather find the integral of. They’ll say the one on the right! The u-substitution one. Although not perfect [2], it’s pretty kewl.

The applets are here:

One     Two     Three     Four     Five

And the applets are dynamic! You can change the lower and upper bounds on the x-f(x) graphs and the lower and upper bounds automatically change on the u-f(u) graph! But because math is awesome, the areas are preserved!

Some things I maybe would have done with the applets in my class:

  • Let kids play with the applets and get familiar with them.
  • For the first applet (starting simple), have kids count the boxes and estimate the area on one graph, and then do it on the other (careful though! the gridlines are different on the two graphs!). Whoa, they are always the same!
  • For the first applet (again, starting simple), ask them to drag the upper limit to the left of the lower limit. Explain what happens and why.
  • The second applet is my favorite! Put the lower limit at x=0. Drag the upper limit to the right. Explain what is happening graphically — and that tie that graphically understanding to the particular u-substitution chosen.
  • In the second applet, can students find three different sets of bounds which give a signed area of 0?
  • In the fourth applet, have students put the lower and upper bounds on x=6 and x=7. Have them calculate the average height of that function in that interval (the area is given!). Do they have visual confirmation of this average height for this interval?Now Looking at the u-graph, the bounds are now u=8 and u=10. Have them estimate the average height of that function in that interval (again, the area is given)! (The average height “halves” in order to compensate for the wider interval. It has to since the areas must be the same) Have students do this again for any lower and upper bounds for this graph. It will always work!
  • In the fifth applet, have students put the lower bound at x=0, and have them drag the upper bound to the right. What can they conclude about the areas of each of the pink regions on the x-f(x) graph? (Alternatively, you can ask: you can see from the u-f(u) graph that the signed area on the original graph will never get bigger than 1, no matter what bounds you choose. Try it! It is impossible! Armed with that information, can you conclude about the pink regions in original graph?)

I’m confident I had more ideas about how to use these when I made them [3]. But it was over a year ago and I haven’t really thought of them since. But anyway, I hope they are of some use to you. Even if you just show them to your kids cursorily to illustrate what graphically is going on when you are doing u-substitution. 


[1] Though I bet if you asked a class why they can use “substitution” when solving a system of equations, what the reasoning is behind this method, they might draw a bit of a blank… But that’s neither here nor there…

[2] What would actually be perfect would be a copy of individual Riemann Sum rectangles from the x-f(x) graph “leaving” the first graph, then in front of the viewer stretching/shrinking their height and width for the appropriate u-f(u) graph, and then floating over to the u-f(u) graph and placing itself at the appropriate place on the u axis. And then a second rectangle does that. And a third. And a fourth. You get the picture. But even though the height and width morph, the area of the original rectangle and the area of the new rectangle will be the same (or to be technical, very very close to the same, since we’re just doing approximations). In this sort of applet, you’d see the actual morphing. That’s what is hidden in my applets above. But that’s actually where the magic happens!

[3] I recall now I was going to make kids do some stuff by hand. For example: before they use the applets, kids would be given lower and upper x-bounds, and asked to calculate lower and upper u bounds. And then use the applets to confirm. Similarly, given lower and upper u-bounds, calculate lower and upper x-bounds. Use the applets to confirm.



    1. I don’t. But I really love the explanation given at the start of Calculus Made Easy (http://www.gutenberg.org/files/33283/33283-pdf.pdf — chapter 4) which goes something like this:

      Let’s see if I can recall it from memory!

      This is just saying that if we increase x by a little bit (dx), then u will increase by some small bit (du).


      Since we have u=1+x^2, we can simplify the above equation to:

      Now I handwave and say (dx)^2 is so small it’s a tinnnnny number squared. Like if dx=0.001, then (dx)^2 would be impossibly small and not really matter. So it really doesn’t matter.


      1. May I interrupt? This reminds me The Analyst, the furious mockery of Newton’s method of fluxions (or maybe Halley?) by Berkeley, you know, vanishing quantities and all that jazz.
        On a serious note, I abenjoyed your applets

  1. I wish there were a way to do it in one graph rather than 2. In my head it would work with a scaling of the x-axis. Like taking the f(x) function and somehow morphing it into the f(u) function by scaling the x-axis via u(x) similar to how x(t) = 2t stretches the t-number line by a factor of 2. Or maybe like watching the videos of topological isomorphisms.

    Then you could see that it’s actually the same function and notice how u(x) transforms everything and where the old x-values ended up. The area would “stay the same object,” but the scaled transformation would be a more geometric object whose area would depend on u?

    I don’t know if it would work or how to make it, but I can kinda visualize it in my head. Not sure if that makes sense.

    1. It totally makes sense. It probably is easily doable (for someone with more skill than me) for simple transformations. For more complicated functions the morphing of space gets all weird looking. I would love to see someone make it!

  2. Not that this is exactly apropos, but I am still somewhat perturbed that I did not get a firm understanding that integration by parts is just the product rule for derivatives in the other direction (see, for example, http://math.ucsd.edu/~wgarner/math20b/int_by_parts.htm). (I can’t swear that this was never taught, but I am pretty sure I never learned it until about 20 years after the first time I integrated by parts. And I still have no idea why the penny finally dropped on the connection; it’s not like I was teaching calculus either.)

  3. As I see it one of the main purposes of studying techniques of integration is to help the development of pattern recognition, so in your example, if students have had some variety of experience in using the chain rule for differentiation they might be able to spot that 2x(1+x^2) is of the form f ‘(u)u’ or df/du times du/dx where u=1+x^2 (in words, the bit on the outside is the derivative of the bit on the inside).
    Then the integral is u^2/2, which is ((1+x^2)^2)/2.
    Substitution, or change of variable, becomes more useful and interesting with 1/(1+x^2) for example.
    Does anybody do the formal definition of ln(x) as integral 1 to x of 1/x ?

  4. Hi Sam,

    I am from the University of Illinois and I thought this was a great way to teach Chain Rule! Giving students a visual way to learn u-substitution gives more meaning to this concepts for the students as they won’t think of it as another boring formula they must memorize but rather a concept they dove deep in which truly enriched their mathematical thinking!

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