Section 3.3 Integration Review
In Analysis 1 you have shown that the Riemann integral
\begin{equation*}
\int_a^b f(t)\, dt
\end{equation*}
is defined for all \(f \in C^0([a,b])\text{,}\) and indeed for somewhat more general functions. Here we recall some basic properties of this integral that we will need in the next section.
Theorem 3.7. Properties of the Riemann integral.
Let \(f,g \in C^0([a,b])\text{,}\) \(\alpha,\beta \in \R\text{,}\) and \(c \in (a,b)\text{.}\) Then
\begin{align*}
\int_a^b (\alpha f(t)+\beta g(t))\, dt \amp=
\alpha \int_a^b f(t)\, dt
+ \beta \int_a^b g(t)\, dt,\\
\int_a^b f(t)\, dt \amp= \int_a^c f(t)\, dt + \int_c^b f(t)\, dt\text{.}
\end{align*}
Moreover, if \(f(t) \le g(t)\) for all \(t \in [a,b]\text{,}\) then
\begin{equation*}
\int_a^b f(t)\, dt
\le \int_a^b g(t)\, dt \text{.}
\end{equation*}
Corollary 3.8. Estimating integrals.
Let \(f \in C^0([a,b])\text{.}\) Then
\begin{equation}
\left\vert \int_a^b f(t)\, dt \right\vert
\le \int_a^b \abs{f(t)}\, dt\tag{3.3}
\end{equation}
and
\begin{equation}
(b-a)\inf_{t \in [a,b]} f(t)
\le
\int_a^b f(t)\, dt
\le (b-a)\sup_{t \in [a,b]} f(t)\text{.}\tag{3.4}
\end{equation}
Proof.
The first inequality
(3.3) follows from
Theorem 3.7 and the fact that
\(-\abs{f(t)} \le f(t) \le \abs{f(t)}\) for all
\(t \in [a,b]\text{.}\) The second inequality
(3.4) similarly follows from the inequalities
\begin{equation*}
\inf_{t \in [a,b]} f(t)
\le f(t) \le
\sup_{t \in [a,b]} f(t)
\end{equation*}
for all \(t \in [a,b]\) combined with the fact that \(\int_a^b 1\,dt =
b-a\text{.}\)
Lemma 3.9. Inertia principle for integrals.
Let \(f \in C^0([a,b])\) and suppose that \(f(t) \ge 0\) for all \(t\in
[a,b]\text{.}\) If \(\int_a^b f(t)\, dt = 0\text{,}\) then \(f(t) = 0\) for all \(t
\in [a,b]\text{.}\)
Proof.
Suppose for the sake of contradiction that there exists some \(c \in
(a,b)\) such that \(f(c) \gt 0\text{.}\) Since \(f\) is continuous, we can find \(\delta \gt 0\) such that for any \(t \in [a,b]\) with \(\abs{t-c} \lt
\delta\) we have
\begin{equation*}
\abs{f(t) - f(c)} \lt \frac 12 f(c)
\end{equation*}
and hence
\begin{equation*}
f(t) \gt \frac 12 f(c).
\end{equation*}
\begin{equation*}
\int_a^b f(t)\, dt
=
\int_a^{c-\delta} f(t)\, dt
+ \int_{c-\delta}^{c+\delta} f(t)\, dt
+ \int_{c+\delta}^b f(t)\, dt,
\end{equation*}
where we have assumed that \(\delta \gt 0\) is chosen small enough that \(c \pm
\delta \in (a,b)\text{.}\) Since by assumption \(f(t) \ge 0\) for all \(t \in [a,b]\text{,}\) while \(f(t) \ge \frac 12 f(c)\) for \(t\) in the smaller interval \([c-\delta,c+\delta]\text{,}\) we can then estimate
\begin{gather*}
\int_a^b f(t)\, dt
\ge
\int_{c-\delta}^{c+\delta} f(t)\, dt
\ge
2\delta \frac{f(c)}2 \gt 0,
\end{gather*}
contradicting the choice of \(c\text{.}\) The cases when \(c\) is one of the endpoints \(a,b\) are extremely similar, and so we omit them.
Finally, we remark that not all functions \(f \maps [a,b] \to \R\) have well-defined definite integrals, indeed, not even all bounded functions. This is discussed in great detail in the Measure theory & integration unit.
Exercises Exercises
1. (PS6) Integrals involving piecewise-linear functions.
For any \(c \in (0,1]\text{,}\) let \(g_c \in C^0([-1,1])\) be the function defined by
\begin{equation*}
g_c(t) =
\begin{cases}
0 \amp \text{if } {-1} \le t \le 0, \\
t/c \amp \text{if } 0 \lt t \le c, \\
1 \amp \text{if } c \lt t \le 1
\end{cases}\text{.}
\end{equation*}
(a)
Draw a graph of \(g_c\) and calculate \(\int_{-1}^1 g_c(t)\, dt\text{.}\)
(b)
For \(d \in [c,1]\text{,}\) consider the function \(f \in C^0([-1,1])\) defined by
\begin{equation*}
f(t) = g_c(t)-g_d(t).
\end{equation*}
Argue that \(0 \le f(t) \le 1\) for all \(t \in [-1,1]\) and that \(f(t)=0\) for \(t \notin [0,d]\text{.}\)
Hint.
I strongly recommend sketching the graph of \(f\text{.}\)
(c)
Conclude that the function
\(f\) defined in
Part b satisfies the inequalities
\begin{align*}
0 \amp \le \int_{-1}^1 f(t)\, dt \le d,\\
0 \amp \le \int_{-1}^1 (f(t))^2\, dt \le d\text{.}
\end{align*}
Do not just calculate the integrals explicitly. In addition to being tedious and slow, this will not help you learn the relevant analysis skills.
Hint 1.
Seriously, do not explicitly calculate the integrals. Instead, take advantage of the inequalities you have already shown in the previous part.
Hint 2.
