I'm trying to understand the importance of the fundamental theorem of calculus, by trying to figure out what question it answers and what new information it provides ( or, what we would " lose" in case we did not have this theorem at hand). My question is simply a conceptual one, it does not aim at any kind of technical accuracy.

Would it be correct to represent what is going on here in the following way ?

(1) for some reason , we are interested in determining what is the area under the curve representing a function f ( because this area is identical to some quantity we are interested in, say Power in physics, or maybe, some probability).

(2) we know how to define this area as an integral : the limit of a sum ( of small rectangles the number of which tends to infinity).

(3) the problem is that calculating this limit of an infinite sum is extremely difficult ( maybe we have no formula to compute this limit of an infinite sum?)

(4) fortunately , we know how to practice anti-differentiation ( prevously defined independently of integration)

(5) and at this point, the FTC tells us that integrating function f is the same thing as finding its antiderivative, or primitive ( which we are perfectly able to do).

Hence my question : can one say that the major interest of the FTC is that is allows us to find " riemann sums" indirectly without having to do all the difficult calculations that would be necessary to compute these sums ( that is, these integrals). Is point (3) above correct?

  • 2
    $\begingroup$ Yes, point $3$ is certainly correct. $\endgroup$
    – lulu
    Feb 2 '20 at 14:10
  • 4
    $\begingroup$ What we get from FTC is not just (3), but also we see that every continuous function on an interval has an antiderivative: integrate the function from a fixed point to a variable endpoint. For example, $e^{-x^2/2}$ on the real line has an antiderivative, such as $\int_0^x e^{-t^2/2}\,dt$ or $\int_{5.32}^x e^{-t^2/2}\,dt$ or $\int_{-\infty}^x e^{-t^2/2}\,dt$. These are not "elementary" functions of $x$, but they are meaningful. Without the construction of definite integrals how would you know all continuous functions on an interval have an antiderivative on that interval? $\endgroup$
    – KCd
    Feb 2 '20 at 14:45

Short answer: The FTC says that anti-differentiation is "the same" as computing areas under functions.

For the beginner, that gives a cool way to compute areas (which they would find hard to do by numerical summation methods), because for their functions, they know other methods how to anti-differentiate.

But for the advanced mathematician, that gives a cool way to compute antiderivatives (which they know is impossible symbolically with other methods), because what they can always do is using numerical summation methods.

Longer answer: Let $f$ be a continuous function on some closed interval $[a,b]$. The FTC usually is stated in two parts:

  1. The function $A(x) := \int_a^x f(t) dt$ is continuous on $[a,b]$, and for $x \in (a,b)$ we have $A'(x) =f(x)$.

  2. For any antiderivative $F$ of $f$ (i.e. a function which is continuous on $[a,b]$ and such that $F'(x) = f(x)$ for all $x \in (a,b)$, we have $\int_a^b f(x)dx = F(b)-F(a)$.

Every time I teach a Calculus class I see that the students love the second part and don't like the first part. I remember I was the same. But what happened when I finally understood it is that now I marvel at the first part, whereas the second part seems like an easy corollary to me.

Your combined points of 3,4 make sense of the second part, and that is good for starters. To understand the first part, and its beauty, one has to realise that your point no.4 is valid only in a very narrow sense. Actually, we "know how to anti-differentiate" only for a small class of functions: Namely, powers of $x$, some elementary trigonometric and inverse trigonometric and exponential and logarithmic functions. OK, then one typically learns that many seemingly more complicated functions can actually be reduced to one of those via substitution or one of the fancier integration techniques. That's all good and well and for all that your point 4 is valid and then your point 3 gives the nice use of the second part of FTC.

What cannot be stressed enough, especially to students straight out of a calculus class, is that it's very easy to write down functions for which unfortunately, we do not "know how to practise anti-differentiation" (examples: $\frac{\sin(x)}{x}$; $\frac{1}{\ln(x)}$, cf. KCd's comment); actually, if you write down a random formula for a function, like $$f(x):=\dfrac{\sqrt[7]{x^5-4x^3+tan(x^{1/3}) -1/x}}{\arcsin(19x^{7/3}+1)+10},$$ it would be a real surprise if there is any chance you can write down an antiderivative ...

... so your point 4 falls flat, and point 3 loses its meaning ...

... but the FTC still says something! Namely, now the first part of the FTC kicks in, and there is one way to easily write down an antiderivative. For example,

$$F(x):= \displaystyle \int_{17}^x \dfrac{\sqrt[7]{t^5-4t^3+tan(t^{1/3}) -1/t}}{\arcsin(19t^{7/3}+1)+10}dt$$

is one. And you can compute it e.g. by computing those Riemann sums (or use estimates like the trapezoidal rule or Simpson's rule; and of course a machine can now do those things quickly for you) -- and there you have an antiderivative! And how else would you have gotten it?

  • $\begingroup$ Thanks for this detailed and enlightning answer! $\endgroup$
    – user655689
    Feb 3 '20 at 8:51

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