Calculus – Why Can’t the Second Fundamental Theorem Be Proved in Two Lines?

calculusinfinitesimalsreal-analysis

The second fundamental theorem of calculus states that if $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$ on the same interval, then:
$$\int_a^b f(x) dx= F(b)-F(a).$$

The proof of this theorem in both my textbook and Wikipedia is pretty complex and long. It uses the mean value theorem of integration and the limit of an infinite Riemann summation. But I tried coming up with a proof and it was barely two lines. Here it goes:

Since $F$ is an antiderivative of $f$, we have $\frac{dF}{dx} = f(x)$. Multiplying both sides by $dx$, we obtain $dF = f(x)dx$. Now, $dF$ is just the small change in $F$ and $f(x)dx$ represents the infinitesimal area bounded by the curve and the $x$ axis. So integrating both sides, we arrive at the required result.

First, what is wrong with my proof? And if it is so simple, what is so fundamental about it?

Multiplying the equation by $dx$ should be an obvious step to find infinitesimal area, right? Why is the Wikipedia (and textbook) proof so long?

I have also read that the connection between differential and integral calculus is not obvious, making the fundamental theorem a surprising result. But to me, it seems trivial. So, what were the wrong assumptions I made in the proof and what am I taking for granted?

It should be noted that I have already learnt differential and integral calculus and I am being taught the "fundamental theorem" in the end and not as the first link between the two realms of calculus.

In response to the answers below: If expressing infinitesimals on their own is not "rigorous" enough to be used in a proof, then what more sense do they make when written along with an integral sign, or even in the notation for the derivative? The integral is just the continuous sum of infinitesimals, correct? And the derivative is just the quotient of two. How else should these be defined or intuitively explained? It seems to me that one needs to learn an entirely new part of mathematics before diving into differential or integral calculus. Plus we do this sort of thing in physics all the time.

Best Answer

The problem with your proof is the assertion

Now $dF$ is just the small change in $F$ and $f(x)dx$ represents the infinitesimal area bounded by the curve and the $x$ axis.

That is indeed intuitively clear, and is the essence of the idea behind the fundamental theorem of calculus. It's pretty much what Leibniz said. It may be obvious in retrospect, but it took Leibniz and Newton to realize it (though it was in the mathematical air at the time).

The problem calling that a "proof" is the use of the word "infinitesimal". Just what is an infinitesimal number? Without a formal definition, your proof isn't one.

It took mathematicians several centuries to straighten this out. One way to do that is the long proof with limits of Riemann sums you refer to. Another newer way is to make the idea of an infinitesimal number rigorous enough to justify your argument. That can be done, but it's not easy.

Edit in response to this new part of the question:

Plus we do this sort of thing in physics all the time.

Of course. We do it in mathematics too, because it can be turned into a rigorous argument if necessary. Knowing that, we don't have to write that argument every time, and can rely on our trained intuition. In fact you can safely use that intuition even if you don't personally know or understand how to formalize it.

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