Why does Apostol introduce the $\int_{a}^{b}f(t)dt=F(b)-F(a)$ property at this point (before discussing differentiation)? Motivations?

Question

So far I have only learnt about what definite integrals are, how to compute them, some applications, etc. Now, Apostol has introduced the concept of indefinite integrals (Calculus Vol. 1, pg. 120).

He defines an indefinite integral of some function $f$ as a new function $F$ such that $$F(x)=\int_{n}^{x}f(t)dt,$$ where $n$ is a constant and $x$ is a variable.

He then goes on to describe some properties of indefinite integrals. One property states that for any definite integral of $f$ with lower and upper bounds $a$ & $b$ respectively, the definite integral of $f$ can be computed by evaluating an indefinite integral of $f$ at $a$ & $b$ and taking the difference, i.e, $$\int_{a}^{b}f(t)dt=F(b)-F(a)$$ I understand that this property is true, however what I don't understand is what the purpose of this property is.

Essentially, he says that if we want to evaluate some definite integral of $f$ with bounds $a$ & $b$, what we can do instead is evaluate two different definite integrals of $f$, namely $\int_{n}^{b}f(t)dt$ & $\int_{n}^{a}f(t)dt$, and take their difference.

But why would we ever want to do this? This doesn't seem like a very useful technique for computing definite integrals, unless we knew the equation for $F$ from the start.

I understand that this property becomes very important and useful down the line once derivatives are introduced, but shouldn't this property be introduced then, after learning about derivatives? Why am I learning about it here? In this context, it feels very unmotivated.

So then why did Apostol decide to introduce this property now rather than later?

EDIT: What I am told in the comments of this post (I think) is that all I must know at this point is that the property is true; that there exists some function $F$ such that finding the equation for $F$ essentially means we're able to turn any "evaluate this definite integral of $f$" problem into a basic subtraction problem. Right now we have no technique for finding the equation for $F$, but down the line we will (once we learn about antidifferentiation).

So, at this point, our only job as the reader is to understand that such a function exists, understand how it is that this property is true (by going through the proof), and move on. That's all.

Is this correct?

Answer 1

Once definite integrals have been taught, the existence of primitives (a.k.a. antiderivatives) is a direct consequence of the (very intuitive) additive property of areas. It would be even less justifiable to not formalize the notion at that point. It's not done for the practical purpose of actually evaluating any definite integral, but for the pedagogical purpose of conveying and consolidating the meaning of definite integrals.

Answer 2

In section 1.23, Apostol computes directly from his definition of the integral that $$\int_0^b x^p\,dx = \frac{b^{p+1}}{p+1}.$$

Putting a zero as the lower bound just makes the notation in the proof a bit less cumbersome.

Applying the property you mention, we conclude that $$\int_a^b x^p\, dx = \frac{b^{p+1}}{p+1} - \frac{a^{p+1}}{p+1}.$$

You're right to notice that working on $[0, b]$ is not fundamentally different from working on $[a, b]$ (that's what your theorem is saying in fact!). It's just that sometimes starting the interval at a nice number like $0$ or $1$, depending on the situation, can make the notation nicer, and we don't lose anything from it.

Answer 3

One reason to treat integration first is that it applies to all functions that the learner is familiar with—in particular step functions and the modulus function. Differentiation is more awkward. Often, the beginner, on introduction to calculus via differentiation, will assume that any function that can described can be differentiated, perhaps except for certain points. In fact, it was only well into the 19th century that nowhere-differentiable functions were discovered. From the modern perspective of what functions are, it seems natural to describe a reciprocal pair of operations (more correctly, a single operation corresponding to a family of operations reciprocal to it) by introducing the more generally applicable one first.