What's the connection between the indefinite integral and the definite integral?

I want to understand the connection between the primitive function or antiderivative and the definite integral.

My problem with this is the independent variable called t in the formula for the first part of the Fundamental Theorem of Calculus.

Here's a composite of the answers I've already seen for this question. Because of t I don't understand it:

The primitive is a function $F(x)$ such that

$F′(x) = f(x)$

The first part of the FTC, which is the thing that connects differentiation/antidifferentiation to the definite integral, is this:

$$F(x) = \int_a^x f(t) dt $$ or “$F(x)$ is a primitive function of $f(x)$. The lower bounds a is fixed. The upper bounds $x$ is variable.”

You can write $\int f(x) dx$ as $$\int_a^x f(t) dt + C$$

A complete analysis of this problem is given in Richard Courant’s calculus book (linked below) page 109+.

Some of the sites I’ve seen: http://ia700700.us.archive.org/34/items/DifferentialIntegralCalculusVolI/Courant-DifferentialIntegralCalculusVolI.pdf

http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/ftc.html

web.utk.edu/~wneilson/mathbook.pdf page 153-154

www.physicsforums.com/showthread.php?t=212449&highlight=x+dillema

www.jirka.org/diffyqs/htmlver/diffyqsse3.html

math.stackexchange.com/questions/105937/what-does-integration-do

www.math.hmc.edu/calculus/tutorials/fundamental_thm/

www.intuitive-calculus.com/fundamental-theorem-of-calculus.html

Solution 1:

I'll try to be a succint as possible but answering your doubts as far as I can, but you made too many questions. Try to summarize as far as you can, adressing your main concerns that will clear the secondary ones.

A. I want to understand deep down intuitively the connection between the primitive function or antiderivative and the definite integral. I am already aware of at least this much:

The so-called indefinite integral is not an integral. Integrals can be represented as areas but the indefinite integral has no bounds so is not an area and therefore not an integral.

The indefinite integral, in my opinion, should be called "primitive" to avoid confusions, as many people call it. The idea is that we learn how to find derivatives, and then are told: "Well, but what about the inverse problem: If we have a known function, what function should we differentiate to get it?" And here comes the idea of primitive of a function, or rather, primitives. The primitive of a given function, which we denote by

$$\int{ f(x) }dx = F(x)$$

is a function such that $F'(x) = f(x) \text{ ; } (1)$.

The notation was used by Leibniz to denote a function that satisfied $(1)$, using the arbitrary constant $C$ to denote that the function wasn't unique - rather, there was a family of primitives of a given function $f$, since the derivative of a constant is null. This is simple notation, but it has nothing to do with $\int_a^b f$ in the sense that $\int_a^b f$ is a number representing the limit of an integral sum of $f$ over $I = (a,b)$, and $\displaystyle \int f(x) dx +C $ is representing a function. As you say, $$\int{ f(x) }dx + C = F(x)$$ is not an integral sum, but a symbolysm for the function that satisfies $(1)$.

B. The first part of the FTC, which is the thing that connects differentiation/antidifferentiation to the definite integral, is this: (...) etc.

Here you're getting confused. The FTC states:

Let $f$ be a continuous real-valued function defined on a closed interval $[a, b]$. Let $F$ be the function defined, for all $x$ in $[a, b]$, by

$$F(x) = \int_a^x f(t) dt $$

Then $$F'(x) = f(x)$$

for all $x$ in $[a, b]$.

This, in short, says that $F$ is indeed another primitive of $f$.

A consequence of this is the so called second FTC and a corollary. Since two primitives are only different by a constant, we should have

$$F(x) - \int_a^x f(t) dt = C$$

But then putting, $x = a$ gives

$$F(a) = C$$

which states that

$$F(x) - \int_a^x f(t) dt = F(a)$$

or

$$F(x) - F(a) = \int_a^x f(t) dt$$

Plugging in $b$ as the upper bound gives the famous corollary:

$$F(b) - F(a) = \int_a^b f(t) dt$$

which states that if $F$ is a primitive of $f$, the previous equality holds.

The second FTC says:

Let $f$ be a function defined on a closed interval $[a, b]$ that admits a primitive $F$ on $[a, b]$, i.e.:

$$f(x) = F'(x)$$

If $f$ is integrable on $[a, b]$ then

$$\int_a^b f(x) dx = F(b) - F(a)$$

(And this didn't depend on the continuity of $f$! You can try and plot the integrals dependeing on the upper bound of discountinous functions to see how the integral always "behaves" much better than the integrand.)

The connection between the primitives and the deinite integral is thus: If we know that a function admits a primitive in an interval, we can easily calculate it's definite integral by means of the primitive.

Solution 2:

The definite integral $$\int_a^b f(x)\,dx$$ is a number: specifically, it is the net signed area between the lines $x=a$, $x=b$, $y=0$, and the curve $y=f(x)$.

(It is defined to be a limit, if it exists, of certain finite sums, called Riemann sums; but the sums and limit are designed to try to capture the idea of area; see for example this previous answer).

Now, one can try to compute the integral as a limit of certain finite sums. What the Fundamental Theorem of Calculus, Part II, tells you, is that these computations are intimately (though surprisingly) connected with differentiation.

Let's start by considering a function defined on $[a,b]$, and that has an integral $\int_a^b f(x)\,dx$.

Now, for each number $c$, $a\lt c\lt b$, we can consider just the area between $x=a$ and $x=c$, instead of going all the way to $x=b$; this, of course, is $\int_a^c f(x)\,dx$. For each value of $c$, this is a number. So we have defined a new function: the "domain" of the function is "all numbers between $a$ and $b$", and the rule for the function is: "given $c$ in the domain, the value of the function is the net signed area between $x=a$, $x=c$, $y=0$, and $y=f(x)$." Though complicated to write out, this is a real-valued function of real variable: a rule that assigns to every valid input (numbers $c$ between $a$ and $b$) one and only one output (the net signed area corresponding to $\int_a^c f(x)\,dx$). That is, we have a function $$c\longmapsto \int_a^c f(x)\,dx.$$

Another way of writing this is to call this function $\mathcal{F}$, and call the input $t$ (since it's the variable, but $x$ is already in use), and write: $$\mathcal{F}(t) = \int_a^t f(x)\,dx$$ so that $\mathcal{F}(t)$ tells you the net signed area "up to $t$".

The Fundamental Theorem of Calculus, Part II, tells you that if you do this, and then ask "is $\mathcal{F}(t)$ differentiable?", the answer is "Yes, it is differentiable, and in fact: $$\frac{d}{dt}\mathcal{F}(t) = f(t)."$$ (See for example this answer for a graphical explanation).

This leads, in turn, to the Fundamental Theorem of Calculus, Part I, which says that there is an alternative way to compute the integral. Namely, if you can find any function $\mathcal{F}(x)$ with the property that $\frac{d}{dx}\mathcal{F}(x) = f(x)$ for all $x$ in $[a,b]$, then $$\int_a^b f(x)\,dx = \mathcal{F}(b)-\mathcal{F}(a).$$

That is, the First Fundamental Theorem of Calculus says: "There's a way you can avoid doing all those annoying finite sums and limits: if you can find a function whose derivative is $f$, then you can use that function to compute the integral."

Now, if we can find a function $\mathcal{F}(x)$ whose derivative is $f(x)$ for all $x$, then $\mathcal{F}(x)$ can be used to compute any definite integral of $f(x)$: just plug in the limits, get the integral. Since this is much simpler than trying to do the Riemann sums and the limits, though by no means a trivial task, we are led to the following: instead of trying to do each integral we are faced by doing limits of Riemann sums, we will instead try to do the following:

Given a continuous function $f(x)$, find a function $\mathcal{F}(x)$ such that $\frac{d}{dx}\mathcal{F}(x) = f(x)$ for all $x$.

We call the function $\mathcal{F}(x)$ an "antiderivative of $f(x)$".

So now, instead of "find the net signed area between $x=a$, $x=b$, $y=0$, and $y=f(x)$", we are faced with the task "find an antiderivative of $f(x)$."

(Note that while there is one and only one value for the area, there are many antiderivatives; however, the Constant Function Theorem tells us that if $\mathcal{F}(x)$ and $\mathcal{G}(x)$ are both antiderivatives of $f(x)$, then $\mathcal{F}(x) - \mathcal{G}(x)$ is constant; that means that if we can find one antiderivative $\mathcal{F}(x)$, then any other antiderivative will be of the form $\mathcal{F}(x)+C$, where $C$ is a constant).

Just like "the net signed area between $x=a$, $x=b$, $y=0$, and $y=f(x)$" is abbreviated with a convenient symbol, $$\int_a^b f(x)\,dx$$ (which has some nice notational properties, such as additivity, etc.), we also want some convenient symbol for

"the general antiderivative of the function $f(x)$."

Why? Because (i) we don't want to have to say "the general antiderivative of the function $f(x)$" all the time (we're lazy); (ii) because good notation is suggestive, and if we can find a good notation that has some nice notational properties, that will help in solving problems down the road.

Since "a general antiderivative of the function $f(x)$" (call it $\mathcal{F}(x)$) can be used to find any definite integral via the Fundamental Theorem of Calculus, connecting the notion to the definite integral is a good idea (though this is the main use of antiderivatives students encounter early on, they turn out to be important for other things, particularly to solve differential equations that turn up when we describe physical phenomena all the time). And so, since $\mathcal{F}(x)$ is useful to compute $$\int_{\square}^{\triangle} f(x)\,dx$$ for any $\square$ and $\triangle$, we leave those spaces "blank" to suggest "you can use this for any integral". This leads to the notation $$\int f(x)\,dx$$ for an antiderivative of $f(x)$. This leads to a bit of a problem, though: since there is no single answer (many different antiderivatives), we don't want the same symbol to represent different things. So we use the symbol above to represent any and all antiderivatives of $f(x)$ (or "the most general antiderivative of $f(x)$"). So when we write something like $$\int x\,dx = \frac{1}{2}x^2 + C$$ we are saying: "the antiderivatives of $f(x)=x$ are the functions $\mathcal{F}(x) = \frac{1}{2}x^2 + C$ with $C$ constant."