Fundamental Theorem of Calculus

This is my 500th post. I started writing something off the cuff about where I was and how far I’ve come as a teacher since I started blogging (which is when I started teaching). But then I realized that in the past two years (I’ve been teaching for four), I’ve stagnated in my evolution, and I got all depressed and wrote something that would probably resonate some of you, but also that would elicit pity comments. And I those depress me more.

So instead (DEFLECTION ALERT!), I thought I’d post something I came up with last year to deal with the Fundamental Theorem of Calculus, Part I.

Although it is easy enough to tell kids “this is what is says, this is how you apply it” (and they can do it), what I have always had a problem with is explaining: this is what the FTC PT I means, and this is why it works. [1] The reason? It’s freakin’ scary! There are lots of variables being thrown around… I thought I posted it last year after writing it, but apparently I did not.

I came up with a guided worksheet that breaks down the ideas into individual pieces, and helps students work through it:

Excuse the fact that it keeps on referring to FTC Part II… I always conflate which is which.

Regardless, I was pleased at how much better my kids last year understood the theorem. They understood the idea of the dummy variable. And that the integral was simply giving an accumulated area from some starting value to some indefinite, variable value in the future.

I’m hopefully going to start this tomorrow, so keep your fingers crossed.

[1] The cop out way is to only explain: “it’s the derivative of an integral, and since they undo each other, you’re left with the original function.” I feel doing so elides the mechanics of what’s going on. It’s a surface-y (and useful to some degree) way to think about it, but it lacks depth.



  1. Just my personal opinion, but I would almost completely invert the order in which you present these sheets, especially doing the graphical / calculator exploration first. The more students who can figure out FTC on their own, the better, and you can follow up with the stuff on dummy variables for those who need it.

    As it is, you’re telling them something really important on page 1 then giving them 4 more pages of more exploratory work to convince them it’s right and important. Why not give them the exploratory work first, and hit ’em with the upshot at the end?

    1. Hi Bowen,

      Great suggestion.I teach two calculus sections — I might try inverting the presentation in one of them to see how it goes.

      I do wonder if the graphical/calculator exploration will work. My initial impression is that it won’t. Not because they won’t be able to type it into their calculators, but because they won’t know what they’re typing or why. Or how to calculate A(x), since x is a variable in the limit of integration. My kids are not strong with variables other than x — and if they see \int \theta d\theta, there is often confusion and palpitations.

      I don’t think the FTC is hard for them — it’s the symbols and interpreting the symbols. Now that I write this, I see that the real thing I need to do — before introducing the FTC, is have students investigate \int_1^x f(t) dt alone. Talk about it being a function based on x, talk about when it is increasing and decreasing. All that good stuff. Have them do a calculator / graphing exploration. And THEN finally once we’re done with that, introduce FTC. ARGH! I wish I had thought of this yesterday.

      Oh well. Today we’re going to investigate if we believe that \int x^r dx is an anomaly for r=-1 (because it gives a natural log, while all the others give a polynomial (or polynomial-esque) thing… Or if it fits in with the numerical pattern, even though the form it takes is so different from the others… We might not get through that. So, maybe I will have a chance!

      1. Sweet beans on the area under x^{-1}. If you have our Precalc book you can find some ideas on this in the very last investigation of the book.

        Specifically, if f(x) is the area under y = r^{-1} from 1 to x, you get magic stuff: the area from 1 to 2 can be expanded to fit exactly into the space from 3 to 6… in other words

        f(2) + f(3) = f(6)

        Hee hee. Then you can try to find the base of the logarithm by figuring out, approximately, what value of x makes f(x) = 1.

      1. Yeah, I agree with Bowen, too on reversing the order. I had to do a similar investigation into the use of variables in my calculus classes. I started with basics of things like f(x) = 3x +1, so what is f(3)? Why? (Replace x with 3 on one side, so you have to do it on the other side–just like all algebra stuff.) I did that until they got bored and begged to moved on to sillier things where I put up crazy looking stuff where the x was almost hidden in there (“Don’t worry if you don’t understand the math of what I’m writing with sigmas and gradients and other fancy looking variables, just replace the x with the number.”) Then, finally, I threw in some of the integral stuff and transitioned from there. If that’s the hurdle, then it’s a procedural one.

        I also like to start with a more piecewise linear function than the quadratic you used for the graphic section. One where they can find the integral using geometric area formulas (rectangles, triangles, trapezoids, toss in some semi-circles, etc.). I even have them sketch a basic concept of what your A(x) would look like on the same axes as the original (just looking for going up or down and a little bit of curvature, but not much).

  2. At some point you got me thinking about how best to explain the fundamental theorem. I think you asked how to explain that an antiderivative is the opposite of a derivative.

    I thought about it. It’s definitely hard to come up with a good explanation. The only avenue of attack I could come up with is to try to make it a very natural result. I think it becomes a very natural result once you’ve planted two notions. 1) the idea of a taylor expansion type approximation of a function and 2) the notion of taking a discrete approximation to the continuous limit.

    Imagine you are sitting on some function. Some crazy function, lets take for example the elevation of all points on earth. Its messy, its complicated. Heck if I could write down its analytic form.

    But, I have the privilege of being able to look at this function on a very small scale. Small compared to its natural variation length. When I go outside, the ground at my feet seems locally like a plane. Especially here in Ithaca, it is certainly hilly, but even when I am on a hill, beneath my feet the ground forms a plane. At this level, if there were a very very dense fog, foggy enough that I could barely see my hand in front of my face, and you asked me to try to estimate the elevation some 100 yards in front of me, the very natural thing to do would be to measure the angle the ground made at my feet and try to linearly extrapolate that forward. The Taylor expansion is born, and makes sense. Any function, whatsoever, if I want to try to predict its value some small distance away from a point, I need only measure the tangent of the function.
    f(x+e) ~ f(x) + df/dx e

    Lets say I wanted to get the elevation a mile away, its clear that if I just repeated this procedure every 10 feet or so, I’d do well.

    I think this makes it clear that estimating a function is just a matter of summing its derivatives, and we end up very naturally with
    f(x2) = f(x1) + int_{x1}^{x2} dx df/dx

    The fundamental theorem of Calculus.

    Then again, perhaps I’ve just gone around in a circle, but I’m not sure. It’s the best I could come up with.

  3. My personal “ah ha” moment with the FTC came when I realized that it says that its geometric interpretation is that “The rate of change of the area under a function is the value of the function.”

    This seems intuitively true. For example, if a function has a value of 4 then, whatever its history, its area at that point is changing at the rate of 4 square units per linear unit of horizontal increase.

    It also seems like an almost literal reading of the second, “Or alternatively…” version of the FTC that you provide,

    It also helped me understand why the lower bound of the integral, “a”, is arbitrary. It doesn’t matter how much area you’ve accumulated up to a point. The FTC is only talking about the rate of change in area at that point.

  4. Thanks for posting that integration worksheet – I am definitely going to use it when I start integrals in a couple of weeks! Also, congratulations on making it to 500 =)

  5. I apologize in advance for failing to find anything to improve and once again only finding something positive to say, but… my students and I always love the comical asides in your worksheets.

    This latest one (“’Mr. Shah,’ you cry out, “THERE ARE TOO MANY VARIABLES!”) is typically priceless.

    Right up there in the pantheon of Sam J. Shah expressions along with, “Check yo’self!”

    – Elizabeth (aka @cheesemonkeysf on Twitter)

  6. Wow, congratulations on 500 posts. I just found you fairly recently, so I guess I’m a bit late to the party. But here’s hoping the second half will be even better than the first.

  7. I’ve found it hard to get students to remember that the real meaning of a definite integral is area under the curve. They want it to mean anti-derivative, the same way the indefinite integral does. And that makes a hard to see proof harder.

    My first time teaching calculus, I thought I understood the fundamental theorem and could explain it, and when I got to class … I just couldn’t. Terribly embarrassing. So I searched (before the internet) far and wide for a good explanation. I found one I liked. It used a different notation, to avoid the confusion about the meaning of the symbols. I couldn’t figure out how to get the diagram in, but it’s pretty plain – I think you can figure it out. I put it in google docs.

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