Don't see the point of the Fundamental Theorem of Calculus.

$$\frac{d}{dx}\int_a^xf(t)\,dt$$

I would love to to understand what exactly is the point of FTC. I'm not interested in mechanically churning out solutions to problems. It doesn't state anything that isn't already known. Prior to reading about FTC, the integral is defined as the anti-derivative. So, it's basically an operator. "Take the anti-derivative by figuring out whose derivative this is!" Simple. So, what is so "fundamental" about redundantly restating the very definition of the integral? (The derivative of the anti-derivative is the function). This to me is like saying $-(-1) = +1$. Not exactly earth shattering.

Am I missing something with regard to the indefinite vs. definite integral?

If we look at a simple example, $$\frac{d}{dx}\int_1^xt^2 \, dt = \cdots =x^2$$

Can we discuss what exactly this is representing?

1. Why would you even write this? Why would you take the rate of change of an area under the curve? Why would you want to take the derivative of an integral? Or, is this just done to prove something else? When would you even come across this situation in Math? Taking the rate of change of the area under a curve and/or total displacement? (derivative of the definite integral)
2. Also, what is the significance of using $t$ as a variable?
3. Why would you integrate from a constant to a function in the first place? (take area under the curve or compute total displacement)

I don't understand what exactly things FTC even allows anyone to do. Without FTC, I can already evaluate definite integrals. Without FTC, I can already take derivatives. So, with FTC, I can take an integral then take a derivative? So, what's even the point of FTC? I really don't see anything "fundamental" whatsoever about this redundant self-evident "theorem". This is like taking the inverse of an inverse. Right back to f(x), but that's simply a "neat trick" vs. a "Fundamental Theorem of Algebra".

Solution 1:

I am guessing that you have been taught that an integral is an antiderivative, and in these terms your complaint is completely justified: this makes the FTC a triviality.

However the "proper" definition of an integral is quite different from this and is based upon Riemann sums. Too long to explain here but there will be many references online.

Something else you might like to think about however. The way you have been taught makes it obvious that an integral is the opposite of a derivative. But then, if the integral is the opposite of a derivative, this makes it extremely non-obvious that the integral can be used to calculate areas!

Comment: to keep the real experts happy, replace "the proper definition" by "one of the proper definitions" in my second sentence.

Solution 2:

You seem to think that you already know that definite integrals have something to do with antidifferentiation. Probably you think this because $\int_a^b f(x) \, dx$ looks remarkably similar to $\int f(x) \, dx$.

But, without the FTC, these two things have nothing whatsoever to do with one another.

They are two completely unrelated operations which for some bizarre reason share a symbol.

$\int f(x) \, dx$, as you note, means the antiderivative of $f(x)$.

But $\int_a^b f(x) \, dx$ means the area between the curve you get when you graph $f(x)$ and the $x$-axis, over the interval $[a,b]$. Without the FTC, there is no reason to expect this to have anything to do with the antiderivative (or "indefinite integral".)

Solution 3:

As integrals and derivatives are presented in Apostol's Calculus, it becomes quite evident that the relationship between them--the Fundamental Theorem of Calculus--is quite remarkable and a bit unexpected.

Apostol actually introduces the notion of an integral first: the notation $$\int_{x=a}^b f(x) \, dx$$ is intended to represent the signed area enclosed by a function $f(x)$ and the $x$-axis, on the interval $x \in [a,b]$. This idea of "area" is something familiar to us from elementary geometry, and it is not difficult to conceptualize the "area under a curve" as an extension of the areas of more familiar geometric shapes, such as polygons and circles. Thus it seems natural to talk about the area enclosed by the curve of a parabola $f(x) = x^2$ and the $x$-axis on the interval $[0,1]$. Indeed, Archimedes of Syracuse, thousands of years ago, used a method remarkably similar to Riemann sums to obtain areas enclosed by parabolic segments.

Now let's switch gears and talk about derivatives: a derivative $f'(x)$ of a function $f(x)$ at a point $x = a$ has the geometric interpretation of the slope of the tangent line to the function at that point. Loosely speaking, the greater this value, the more rapidly the function $f(x)$ is increasing at that point. More formally, $$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}.$$

What makes the integral and derivative concepts in calculus (or analysis, if you prefer), is that both are mathematical ideas involving some kind of limiting process: the (Riemann) integral is understood as the sum of the rectangular areas defined by successively more refined partitions of the interval $[a,b]$, and the derivative is understood as the slope of a secant line as one intersection point approaches the other.

Note that in these contexts, it is not at all obvious that the two concepts are related. Yet the Fundamental Theorem of Calculus states (in one form) that $$\int_{x=a}^b f(x) \, dx = F(b) - F(a)$$ where $F(x)$ is some function satisfying $F'(x) = f(x)$. This gives us a means to compute without resorting to Riemann summation a definite integral as the difference of the integrand's antiderivative on the interval's endpoints. (In fact, the FTC is a unidimensional special case of Stokes' Theorem and as such holds deeper insights, but that's not in the scope of our discussion).

So, in summary, the FTC is not a trivial result. Apostol does in fact provide a quasi-geometric heuristic "proof" of why this relationship should exist, and it is worth reading. And if we are to have a proper appreciation for calculus, it helps to have the proper pedagogy and motivation that his text provides. But should you desire to understand the foundations of calculus further, then a more rigorous and less computationally oriented treatment is recommended, such that found in Walter Rudin's Principles of Mathematical Analysis.