Reprise on Integration

Recently, I posted a bit asking people how they introduce integrals. And I got a ton of different responses, which was wonderful. I am going to copy a few bits of comments here, but I really recommend that if you teach calculus, you take a moment to read them all in their entirety.

David P.: I sometimes use the physics of displacement/velocity/acceleration to introduce antiderivatives. […] I am only in my 3rd year teaching, so I’ve not found a “best” way yet. I really like the “surprise” of the FTC that areas and slopes seem like they should have no connection whatsoever, but that they’re almost as related at + and -. So, sometimes I even just say, “ok, we’re done with that section, let’s move on to something else” and then try to surprise them when we get to the connection.

Andy: I usually teach anti-derivatives as part of my derivatives unit. […] I also show how it applies to the position, velocity, acceleration problems. Then I transition to my integration unit. I don’t even tell them about integration and anti-differentiation being related. I just talk about area and then when we get the the fundamental theorem, I am able to drop the crazy idea on them that integration and differentiation are related. I enjoy their reactions to that.

Nick H.: Generally, I like the ask questions first teach skills later approach.

TwoPi: I usually start off with velocity examples, and in each case link the displacement calculation (or approximation) to the geometry of computing the area under the graph of velocity versus time. So start with constant velocity, then linear velocity (and sometimes nice quick applications involving stopping distances for cars at various initial velocities).

This year I focused on anti-derivatives. On the first day, I just said: the derivative of x^2 is 2x, so the antiderivative of 2x is x^2. That’s all. The rest of the class had students struggle through finding simple antiderivatives (PDF and PDF). On the second day, I gave students a method to solving antiderivatives, a method which builds their intuition (PDF). And on the third day, I had students just practice, practice, practice.

Then I gave them a quiz — 17 questions asking for the antiderivatives of functions from x\sqrt{2x^2+1} to \frac{e^\sqrt{x}}{\sqrt{x}} to \frac{\cos x}{\sin^2 x}. Moreover, I didn’t give partial credit. If a negative sign was missing, or a constant was incorrect, I took off full credit for the problem.

The average grade for both sections was an A-.

So I have to say that my approach this year worked. I’ll deal with u-substitution and all that nonsense later. But the fact is, my students will be able to soon integrate some pretty hard stuff without resorting to u-substitution.

Many of the comments talked about working with position, velocity, and acceleration graphs to start out. I think after I teach the area under curves and Riemann sums, I will go into this topic. Honestly, I was hesitant to start integration with position/velocity/acceleration because anything physics related tends to make my students convulse. They are scared of physics. I wanted to make sure that they didn’t shut down completely before we even start.

(However, I am excited to derive h(t)=\frac{1}{2}g_{const}t^2+v_0 t+h_0 from first principles. I hope to hear lots of oohs and aahs.)



  1. I’m not a practiced Calculus teacher, but I was glad to hear you have your kids do moderately complicated anti-derivatives without u-substitution. Maybe it’s just because I don’t think I ever learned it myself back in the day, and I think my understanding and efficiency were both better off for it.

  2. @Matt E: Last year I didn’t do it, and I was sad that all my students relied on u-substitutions for the easiest of problems. And then I remembered how my calc teacher taught me — make an educated guess and then compensate based on the constants — and I decided to show my students this way of working.

    What’s precious is that today I showed one of my classes the integral sign, and they were all like “WHAT? That’s all that symbol means?” I think it’s because that’s the standard symbol that gets used to show “complicated” math.

  3. I LOVE that I can look at your blog for direction as I’m teaching Calculus for the first time. Just now starting integrals and I was asking myself this same question….looking for the “best” way to introduce. Thank you so much for being such a great resource!

  4. I am so grateful that I found your blog last year and even more so that I found this series of posts on anti-derivatives! You have saved my life, seriously! This is my first year teaching Calc (and the last time I *did* Calc was way back in the late 1980’s and early 1990’s when I was in HS/college!!) But I knew there was no way I wanted to jump in and start with the “rules” like my textbook does. I’d been forming the idea of using intuition, but your blog and, most importantly, your resources were amazing! I used them nearly as-is and tried it out on both my AP Calc classes today. I was seriously stressing about some other stuff going on and on top of that I was not feeling focused about my approach to this important topic. But let me tell you….it couldn’t have gone better in my opinion! I tried to talk as little as possible, gently nudging and occasionally suggesting, but mostly letting the students work through it and they worked like champs!

    Now I just have to figure out the path forward and keep the momentum going tomorrow building on what they’ve already accomplished. Thanks SO much for sharing this (and not putting a price-tag on it…because what you’ve provided is truly priceless!).

    1. YAY!!! I’m so so so happy to hear that. Literally it’s been such a long day and I still have more work to do — but reading your comment just made me feel like less of a bad teacher!!! If you want to check out all my interesting calc posts (from when I first started teaching calculus a decade ago to when I stopped teaching calculus 8 years ago), here they are (so you don’t have to dig through all the other stuff):

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