Definition 134.
A set of points \(\set{t_0, t_1, \ldots, t_n}\) such that \(a = t_0, b = t_n\text{,}\) and \(t_{i-1} \lt t_i\text{,}\) for all \(i\) is called a partition of the closed interval \([a,b]\text{.}\)
Chapter 5 Integration
Unlike with differentiation, we will need a number of auxilliary definitions for beginning integration.
Example 135.
Give some partitions of \([2,4]\) and \([-1,0]\text{.}\) What would the points of a partition of \([0,1]\) be if \(t_i - t_{i-1}\) is the same for all \(i\text{?}\)
Definition 136.
Given \([a,b]\text{,}\) \(n\in\N\text{,}\) and calling \(\frac{b-a}{n} = \Delta t\text{,}\) we may for convenience give the partition \(\set{a, a+\Delta t, a + 2\Delta t, \cdots, b - \Delta t, b}\) the name regular partition \(P_n\) (where \([a,b]\) is implicit).
Definition 137.
Given a partition \(P = \set{t_0, t_1, \ldots, t_n}\) of \([a,b]\text{,}\) let \(y_i\) be the supremum of \(\setof{f(x)}{x\in [t_{i-1}, t_i]}\text{.}\) We say that the sum
\begin{equation*}
U_P(f) = \sum\limits_{i=1}^n y_i (t_i - t_{i-1})
\end{equation*}
is the upper sum for \(P\) of \(f\) on \([a,b]\text{.}\)
Definition 138.
Given a partition \(P = \set{t_0, t_1, \ldots, t_n}\) of \([a,b]\text{,}\) let \(z_i\) be the infimum of \(\setof{f(x)}{x\in [t_{i-1}, t_i]}\text{.}\) We say that the sum
\begin{equation*}
L_P(f) = \sum\limits_{i=1}^n z_i (t_i - t_{i-1})
\end{equation*}
is the lower sum for \(P\) of \(f\) on \([a,b]\text{.}\)
Definition 139.
A bounded function is a function with a bounded image set.
We will assume all functions in this chapter are bounded.
Example 140.
Give some upper and lower sums on some of these partitions with non-constant bounded functions.
Problem 141.
Let \(f(x) = 0\) for each number \(x\in [0,1]\) except \(x=0\text{,}\) and let \(f(0) = 1\text{.}\) Show that \(U_p(f) \gt 0\) for all partitions of \([0,1]\text{,}\) but that for any positive number \(\varepsilon \gt 0\) there is a partition \(P_{\varepsilon}\) such that \(U_{P_\varepsilon}(f) \lt \varepsilon\text{.}\) In addition, fully describe all lower sums of \(f\) on \([0,1]\text{.}\)
Problem 142.
Let \(f(x) = x\text{,}\) and let \(P_n\) be the (regular) partition of \([0,1]\) given by \(\set{0, \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n-1}{n}, 1}\text{.}\) Compute \(U_{P_5}(f)\) and \(L_{P_5}(f)\text{,}\) and then give a formula for \(U_{P_n}(f)\) and \(L_{P_n}(f)\text{.}\)
Problem 143.
Suppose that \(f\) is bounded, with \(m\) a lower bound and \(M\) an upper bound of \(f([a,b])\text{.}\) Show that, for any partition \(P\) of \([a,b]\text{,}\) \(U_P(f) \le M\cdot (b-a)\) and \(L_P(f) \ge m\cdot (b-a)\text{.}\)
Problem 144.
Suppose that \(f\) is bounded on \([a,b]\) and \(P\) is a partition of \([a,b]\text{.}\) Show that \(L_P(f) \le U_P(f)\text{.}\)
Note that as a result of the previous problems that the set of all upper and lower sums of \(f\) is a bounded point set.
Definition 145.
We say that a partition \(Q\) refines (or is a refinement of) a partition \(P\) if \(P\) and \(Q\) are both partitions of \([a,b]\) and \(P\subseteq Q\text{.}\)
Example 146.
Take some partitions from previous examples, and refine them.
Problem 147.
If \(P\) and \(P'\) are both partitions of \([a,b]\text{,}\) there is a partition \(Q\) which is a refinement of both.
Problem 148.
If \(f\) is a bounded function on \([a,b]\) and \(P\) and \(Q\) are partitions of \([a,b]\) such that \(Q\) refines \(P\text{,}\) show that \(L_P(f) \le L_Q(f)\) and \(U_P(f) \ge U_Q(f)\text{.}\)
We are nearly ready to define (definite) integrals.
Definition 149.
With \(f\) a bounded function on \([a,b]\text{,}\) we define the upper integral from \(a\) to \(b\) of \(f\) to be the infimum of the set of all upper sums for \(f\) on \([a,b]\text{.}\) We denote the upper integral by \(\overline{\int_a^b f}\text{.}\)
Definition 150.
With \(f\) a bounded function on \([a,b]\text{,}\) we define the lower integral from \(a\) to \(b\) of \(f\) to be the supremum of the set of all lower sums for \(f\) on \([a,b]\text{.}\) We denote the lower integral by \(\underline{\int_a^b f}\text{.}\)
Example 151.
Why do the upper integral and lower integral always exist in this context?
Problem 152.
With \(f\) as throughout this chapter, show that \(\overline{\int_a^b f} \ge \underline{\int_a^b f}\text{.}\)
Problem 153.
If we let \(f(x)\) be the function such that \(f(x) = 1\) if \(x\in \Q\) but \(f(x) = 0\) otherwise, show that in this case
\begin{equation*}
\overline{\int_0^1 f} \gt \underline{\int_0^1 f}.
\end{equation*}
We can get more interesting conclusions if \(f\) has more properties.
Problem 154.
Suppose that \(f\) is not just bounded, but continuous. Suppose further that \(f(x) \ge 0\) for all \(x\in [a,b]\) and that for some \(z\in [a,b]\text{,}\) \(f(z) \gt 0\text{.}\) Then show that \(\overline{\int_a^b f} \gt 0\text{.}\)
Definition 155.
For \(f\) as above, if \(\overline{\int_a^b f} = \underline{\int_a^b f}\text{,}\) then we say that \(f\) is integrable on \([a,b]\) and call this common value the integral of \(f\) from \(a\) to \(b\text{.}\) We denote the integral, if it exists, by
\begin{equation*}
\int_a^b f \text{ or } \int_a^b f(x)\ dx.
\end{equation*}
Traditionally when this value exists we say that \(f\) is Riemann integrable and the number \(\int_a^b f\) is the Riemann integral. Technically, however, it is the Darboux integral, with Riemann integrals coming from so-called Riemann sums. The two notions can be proved equivalent.
Since we saw above that the lower and upper integrals can be distinct, there exist nonintegrable bounded functions (in this sense). Luckily, there also exist integrable functions.
Example 156.
Show that constant functions \(f(x) = c\) are indeed integrable over any closed interval.
It is worth pondering why the information we have from previous examples is not quite enough to show that we have more integrable functions. After all, readers may be familiar with computing integrals. Historically, though, finding the values of many integrals (originally called quadrature since one wished to construct an actual square with the same area) was quite challenging. Since many applications in a variety of fields come directly from approximations involving the sums we have been computing, it would be best to have exact values for these applications, not just approximations from the sums. Keep this in mind as we progress.
A first step in this direction is the following characterization, which is very useful in showing functions are integrable.
Problem 157.
Let \(f\) be a bounded function on \([a,b]\text{.}\) Show that if \(f\) is integrable then for all \(\varepsilon \gt 0\) there is a partition \(P\) of \([a,b]\) such that \(U_P(f) - L_P(f) \lt \varepsilon\text{,}\) or show the converse (both are true and useful).
Problem 158.
Show that the following function is integrable on \([0,1]\text{,}\) and compute the integral. You may want to use previous problems.
\begin{equation*}
f(x) = \begin{cases} 1 \amp x = 0\\ 0 \amp x\ne 0. \end{cases}
\end{equation*}
This is not a lot of material to integrate. The next sets of theorems will vastly expand our repertoire of functions known to be integrable, though not yet of those with precise values at hand.
Definition 159.
A function is nondecreasing if for each \(x \lt y\) in the domain of \(f\) we have \(f(x) \le f(y)\text{.}\) We say that it is (strictly) increasing if \(f(x) \lt f(y)\) always. The concept of nonincreasing is defined with \(\ge\) in place of \(\le\text{.}\)
Example 160.
Give several nondecreasing functions we haven't already used.
Problem 161.
Prove that a nondecreasing bounded function \(f\) on a closed interval \([a,b]\) is integrable.
You may want to write down a number of upper and lower sums to prepare for the previous problem. What previous problem will likely be easiest to use to prove integration theorems?
Example 162.
Since \(\sqrt{1-x^2}\) is nonincreasing on \([0,1]\text{,}\) this shows that \(4\int_0^1 \sqrt{1-x^2}\, dx\) exists. What number is this?
This example is clever, but still holds a bit of magic, since even a small change from \(\sqrt{1-x^2}\) might make it very hard. Could we integrate in general?
The preceding results mean that power functions, exponentials, and many other standard functions are integrable on certain intervals, even if we can’t finish computing them yet. The next theorems verify that most related functions are integrable on most typical intervals, and will help evaluate them too.
Problem 163.
Let \(c\in\R\) and let \(f\) be integrable on \([a,b]\text{.}\) Show that \(cf\) is integrable on \([a,b]\) and \(\int_a^b cf = c \int_a^b f\text{.}\)
Problem 164.
Let \(f\) and \(g\) be integrable on \([a,b]\text{.}\) Show that \(f+g\) is integrable on \([a,b]\text{.}\) Can you show the formula \(\int_a^b (f+g) = \int_a^b f + \int_a^b g\text{?}\)
Not every possible statement is true, of course.
Problem 165.
Give two functions \(f\) and \(g\) which are integrable on \([0,1]\) such that \(fg\) is integrable on \([0,1]\) but \(\left(\int_0^1 f\right) \left(\int_0^1 g\right) \ne \int_0^1 fg\text{.}\)
We will now head straight for the punchline to the story of calculus, the 'fundamental theorems'. Just as these were not recognized as such until quite some time after Newton and Leibniz, you may want to reevaluate what you think their importance is here.
We begin by proving some facts about the upper integral (they are also true for the lower integral). Remember, all functions are bounded.
Problem 166.
Assume that \([a,b]\) is a closed interval and \(c\in (a,b)\text{.}\) Show that \(\overline{\int_a^c f} + \overline{\int_c^b f} = \overline{\int_a^b f}\text{.}\)
On a related note, for upper (and indeed for lower and Riemann) integrals, we define
\begin{equation*}
\overline{\int_b^a f} = -\overline{\int_a^b f}
\end{equation*}
and
\begin{equation*}
\overline{\int_a^a f} = 0.
\end{equation*}
Can you see why we do this?
Problem 167.
Show that if \(f\le g\) on the interval \([a,b]\text{,}\) then \(\overline{\int_a^b f} \le \overline{\int_a^b g}\text{.}\)
The next property is of course also true for the regular integral (when it exists), and is called the integral mean value theorem. Can you see why?
Problem 168.
Let \(f\) be a continuous function on \([a,b]\text{;}\) then there is a number \(c\in [a,b]\) such that \(\overline{\int_a^b f} = f(c) (b-a)\text{.}\)
Hint.
Use Problem 110 and the previous problem.
The next thing we do is really expand our collection of integrable functions!
Problem 169.
Let \(U\) be defined such that for each \(x\in [a,b]\text{,}\) \(U(x) = \overline{\int_a^x f(t) \, dt}\text{,}\) and let \(L(x) = \underline{\int_a^x f(t) \, dt}\text{.}\) Show that if \(f\) is continuous on \([a,b]\text{,}\) then \(U\) and \(L\) are differentiable at every \(c\in [a,b]\text{,}\) with \(U'(c) = f(c) = L'(c)\text{.}\)
Problem 170.
In Problem 169, show that in fact \(U(x) = L(x)\text{,}\) and conclude that if \(f\) is continuous on \([a,b]\) then \(f\) is integrable on \([a,b]\text{.}\)
And finally we get the most beloved theorems of all. Their true importance lies in the inverse relationship between derivatives and integrals, and that one can evaluate integrals (which a priori come from the sums we did earlier) exactly using information about derivatives. You should be able to prove these without lots of subtle work, using the facts about lower and upper integrals.
Problem 171. Fundamental Theorem of Calculus I.
Let \(f\) be a continuous function on \([a,b]\text{,}\) and let \(F\) be defined such that for each \(x\in [a,b]\text{,}\) \(F(x) = \int_a^x f\text{.}\) Then for each point \(x\in [a,b]\text{,}\) \(F\) has a derivative at \(x\) and we can compute \(F'(x) = f(x)\text{.}\)
Problem 172. Fundamental Theorem of Calculus II.
Suppose that \(g\) is a function on \([a,b]\text{,}\) that \(g\) has a derivative at each point of \([a,b]\text{,}\) and that the function \(g'\) is itself continuous at each point in \([a,b]\text{.}\) Then show that \(\int_a^b g' = g(b) - g(a)\text{.}\)
These are the keys which made the efforts of Newton and Leibniz into the calculus—something that one could calculate with. It is worth noting that Problem 172 is true even if \(f'\) is just integrable—truly, a great theorem.
