Why is differential calculus often presented before integral calculus?

Question

Why is differential calculus often presented before integral calculus?

Note: I'm still learning calculus at the moment.

It seems that many elementary calculus texts describe differential calculus before integral calculus. They start with an informal intuition into the concept of a limit and how to calculate various limits. They then go on to describe the derivative via physical applications and/or the tangent/secant line approach. To me it seems that the integral would be more intuitive to understand first, and then limits and derivatives. To the best of my knowledge, this is also how calculus was historically developed. Is there a particular reason for presenting the derivative before the integral in elementary calculus?

Answer 1

David Bressoud has a 2019 book that argues for teaching integration before derivatives, and only later teaching limits, partly because that matches the way that the ideas developed historically: "Calculus Reordered: A History of the Big Ideas".

Answer 2

While there are no theoretical difficulties with developing integration first ("from scratch" measure theory books demonstrate this), it presents some pedagogical challenges.

Most problems in existing Calculus courses have "closed form" solutions. It is rare to just set up a problem and then numerically evaluate. The primary tool available for exact calculation of definite integrals is the Fundamental Theorem of Calculus. Finding $$\int_0^1 x^5 \textrm{d}x$$ can be done numerically or exactly using excruciating Riemann sum calculations. We like to know that $\frac{d}{dx} x^6 = 6x^5$ before covering integration so that this integral can be evaluated exactly using the fundamental theorem.

I personally think it would be a better story if we taught integration first, got practice setting up all kinds of integrals to solve practical problems, and used numerical approximation as our only evaluation tool. This would drive home what the definite integral IS. Then do differential calculus in all of its glory. Then the FTOC would actually be surprising and useful (Wow! Now we can calculate these integrals exactly instead of just approximating!). As it is, most students think the FTOC is the definition of the definite integral, which is unfortunate.

Answer 3

The text I know of that does integration first is: Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra.

An advantage of using this for top-rate incoming university students is that they do not think "I already know this" and tune out [as they may do when the course begins with differentiation].

Answer 4

Because:

https://xkcd.com/2117/

And now I'm padding my answer with 21 characters.

Answer 5

One issue is that for differentiation, you can find the derivatives of so-called "elementary functions" (e.g. powers, exponentials, logarithms, trig functions) directly from the definition of a limit, and there are then standard ways to find the derivatives of an expression knowing the derivatives of its component parts (e.g. the product and quotient rules, function-of-a-function, etc.)

Therefore it is straightforward to generate a large number of "textbook exercises" of graded difficulty.

On the other hand, most "simple-looking functions" do not have closed-form integrals, and the easiest way to find a set of basic functions that do have closed form integrals is to recognize that "the function you want to integrate looks like a derivative that you already know".

In fact many of the so-called "special functions" in mathematics are defined as the integral of a simple-looking function (e.g. the gamma function, which generalizes the idea of a factorial for non-integer values).

Of course as other answers have said, if you taught integration using only numerical methods as an initial way to evaluate integrals, that problem does not exist, but (at high school or university level) most students will not have much if any understanding of numerical methods, and therefore there are too many possibilities for "garbage-in garbage out" exercises where students have no way to check their work.

Typing formulas into Wolfram's integral evaluator isn't "learning math" - and it isn't even an interesting task, unless you are going to do something interesting with the output.

Answer 6

I think one of the main reasons for the ordering, limits-> derivatives -> integrals is that it follows in order of difficulty in some sense of the word.

In reality the situation is more complex, but if you start with the minimal (read $\epsilon$-$\delta$) definition of a limit it's fairly straightforward. It's also almost essential for even a semi rigorous treatment of both differential and integral calculus.

Differentiation comes pretty much next in complexity. In reality the basic version (assuming stuff you have is almost everywhere $C^\infty$) is actually quite a bit easier than limits, but since (it looks as if) you're using limits to calculate it, it's more complex. At the same time derivatives are important for a lot of other things you want the students to be learning, mainly basic physics, where you get differential equations at every corner.

Integration is by far the hardest both in terms of students understanding and actual computation. Unlike differentiation where (mostly) anything that can be written neatly has a neat (read made up of elementary functions) derivative, most integrals don't actually have a closed form solution. With "most" here meaning many of the ones students can think of.

All in all, this order is in many ways natural, progressing from easier to harder and building on previously learned material. In reality integration both 2d and 3d is a lot more complex than derivatives. Both in terms of the limits employed (you have to take a limit over partitions, so you need a new notion of limit) and the idea that's it's "finding the area under the curve" is actually surprisingly complex since you don't actually know what "area" is until you come up some version of integration.

Answer 7

I'd like to comment on why, indeed, it would be reasonable to present the subject with integrals first... (which is not what is done, nearly universally, I understand).

In particular, I'd argue that the notion of "derivative" is more sophisticated than "area under a curve". This includes the subtlety of "instantaneous rate of change". Yes, people'd been thinking about this a long time (cf. Zeno's paradoxes).

But/and, yes, a fundamental reason for the success of "calculus" as a very basic mathematical gadget is the fundamental theorem of calculus, giving explicit and elementary (indefinite) integrals for many interesting functions. Explicit (elementary) expressions obviously allow many more manipulations and experimentation than subtler "estimates".

For that matter, the first really rigorous results about differential equations used conversion of differential equations to integral equations, and invocation of theorems about compact operators. (See Volterra, Hilbert, Schmidt.)

In my own experience, I did think it was charming that slopes to polynomial curves could be exactly determined, but it was really amazing that areas under such graphs could be precisely (not just numerically) determined, and in forms that were astonishingly simple.

Answer 8

As with a huge number of the "why do we do this order" questions, or the "why don't we teach real analysis before calculus" questions, they seem to tacitly assume that the order of teaching is based on philosophical explanation or even formal mathematical proof. WRONG!

This is NOT the reason why instruction is structured the way it is. It is structured based on imperfect beings that learn harder subjects after easier ones, because that works better pedagogically. Instruction is a maximization problem of time, brains, etc. To get the most sense pounded into skulls within the practical limits that pertain in the practical world. And in the case of integration, it is harder than differentiation (going backwards) and makes more sense afterwards. So. Practical pedagogy uber theoretical explication.

Answer 9

I have come to realize another very practical reason for teaching differentiation before integration.

In most applications of integration we are splitting something (area, volume, arclength, work, etc) into lots of tiny pieces, compute an approximation of each piece, and then sum. Taking the limit as the size of the pieces tends to $0$ yields an integral.

What is underappreciated, I think, is that often the "approximate each piece" step will involve differentiation.

Take arc length as an example.

We have a function $f: [a,b] \to \mathbb{R}$ and we want to calculate the arc length of the graph of $f$.

It is natural to first subdivide the interval $[a,b]$ into $N$ equal sized subintervals $[x_k, x_{k+1}]$ with $x_k = a + \frac{b-a}{N}(k-1)$ for $k=0,1,2, 3 \dots, N, N+1$.

Since arclength is additive, the length of the whole curve is the sum of the length over each subinterval.

We then approximate each small arc by the length of the secant line connecting the two endpoints $\sqrt{(f(x_{k+1}) - f(x_k))^2 + (x_{k+1}-x_k)^2}$.

The issue is that

$$\sum_0^{N-1} \sqrt{(f(x_{k+1}) - f(x_k))^2 + (x_{k+1}-x_k)^2}$$

is not in the form of a Riemann sum for any function.

So we need to approximate each of these summands again using the derivative:

$$ \sqrt{(f(x_{k+1}) - f(x_k))^2 + (x_{k+1}-x_k)^2} \approx \sqrt{(f'(x_k)(x_{k+1}-x_k))^2 + (x_{k+1}-x_k)^2} $$

So we obtain

$$\sum_0^{N-1} \sqrt{(f'(x_k))^2 + 1} \Delta x$$

as our approximation of the arc length.

This is an extremely common phenomenon: forming an integral to compute a quantity of interest relies on decomposition into small pieces, but approximating these small pieces requires differentiation. This is a good mathematical reason for studying differentiation first.