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8.1 Definition of the Integral

If $f$ is a monotonic function from an interval $[a,b]$ to $\mbox{{\bf R}}_{\geq 0}$, then we have shown that for every sequence $\{P_n\}$ of partitions on $[a,b]$ such that $\{\mu (P_n)\}\to 0$, and every sequence $\{S_n\}$ such that for all $n\in\mbox{${\mbox{{\bf Z}}}^{+}$}$ $S_n$ is a sample for $P_n$, we have

\begin{displaymath}\{\sum (f,P_n,S_n)\}\to A_a^bf.\end{displaymath}

8.1   Definition (Integral.) Let $f$ be a bounded function from an interval $[a,b]$ to $\mbox{{\bf R}}$. We say that $f$ is integrable on $[a,b]$ if there is a number $V$ such that for every sequence of partitions $\{P_n\}$ on $[a,b]$ such that $\{\mu (P_n)\}\to 0$, and every sequence $\{S_n\}$ where $S_n$ is a sample for $P_n$

\begin{displaymath}\{\sum (f,P_n,S_n)\}\to V.\end{displaymath}

If $f$ is integrable on $[a,b]$ then the number $V$ just described is denoted by $\displaystyle {\int_a^bf}$ and is called `` the integral from $a$ to $b$ of $f$.'' Notice that by our definition an integrable function is necessarily bounded.

The definition just given is essentially due to Bernhard Riemann(1826-1866), and first appeared around 1860[39, pages 239 ff]. The symbol $\displaystyle {\int}$ was introduced by Leibniz sometime around 1675[15, vol 2, p242]. The symbol is a form of the letter s, standing for sum (in Latin as well as in English.) The practice of attaching the limits $a$ and $b$ to the integral sign was introduced by Joseph Fourier around 1820. Before this time the limits were usually indicated by words.

We can now restate theorems 7.6 and 7.15 as follows:

8.2   Theorem (Monotonic functions are integrable I.) If $f$ is a monotonic function on an interval $[a,b]$ with non-negative values, then $f$ is integrable on $[a,b]$ and

\begin{displaymath}\int_a^bf=A_a^bf=\alpha(\{ (x,y)\colon a\leq x\leq b \mbox{ and } 0\leq y\leq
f(x)\}).\end{displaymath}

8.3   Theorem (Integrals of power functions.) Let $r\in\mbox{{\bf Q}}$, and let $a,b$ be real numbers such that $0<a\leq b$. Let $f_r(x) = x^r$ for $a \leq x \leq b$. Then

\begin{displaymath}\int_a^b f_r = \cases{ \displaystyle {{b^{r+1} - a^{r+1} \ove...
...setminus \{-1\}$\vspace{1ex}\cr
\ln(b) - \ln(a) & if $r=-1$.}
\end{displaymath}

In general integrable functions may take negative as well as positive values and in these cases $\displaystyle {\int_a^bf}$ does not represent an area.

The next theorem shows that monotonic functions are integrable even if they take on negative values.

8.4   Example (Monotonic functions are integrable II.) Let $f$ be a
monotonic function from an interval $[a,b]$ to $\mbox{{\bf R}}$. Let $B$ be a non-positive number such that $f(x)\geq B$ for all $x\in [a,b]$. Let $g(x)=f(x)-B$.
\psfig{file=ch8a.eps,width=2in}
Then $g$ is a monotonic function from $[a,b]$ to $\mbox{{\bf R}}_{\geq 0}$. Hence by theorem 7.6, $g$ is integrable on $[a,b]$ and $\displaystyle {\int_a^bg=A_a^b(g)}$. Now let $\{P_n\}$ be a sequence of partitions of $[a,b]$ such that $\{\mu (P_n)\}\to 0$, and let $\{S_n\}$ be a sequence such that for each $n$ in $\mbox{${\mbox{{\bf Z}}}^{+}$}$, $S_n$ is a sample for $P_n$. Then
\begin{displaymath}
\{\sum (g,P_n,S_n)\}\to A_a^b(g).
\end{displaymath} (8.5)

If $P_n=\{x_0,\cdots ,x_m\}$ and $S_n=\{s_1,\cdots
,s_m\}$ then

\begin{eqnarray*}
\sum (g,P_n,S_n) &=& \sum_{i=1}^m g(s_i)(x_i-x_{i-1})\\
&=& \...
...-1}) - B\sum_{i=1}^m(x_i-x_{i-1})\\
&=&\sum (f,P_n,S_n)-B(b-a).
\end{eqnarray*}



Thus by (8.5)

\begin{displaymath}\{\sum (f,P_n,S_n)-B(b-a)\}\to A_a^b(g).\end{displaymath}

If we use the fact that $\{B(b-a)\}\to B(b-a)$, and then use the sum theorem for limits of sequences, we get

\begin{displaymath}\{\sum (f,P_n,S_n)\}\to A_a^b (g)+B(b-a).\end{displaymath}

It follows from the definition of integrable functions that $f$ is integrable on $[a,b]$ and

\begin{displaymath}\int_a^bf=A_a^b(g)+B(b-a)=\int_a^bg+B(b-a)=\int_a^bg-\vert B\vert(b-a).\end{displaymath}

\psfig{file=ch8ba.eps,width=1.25in} \psfig{file=ch8bb.eps,width=1.25in} \psfig{file=ch8bc.eps,width=1.25in} \psfig{file=ch8bd.eps,width=1.25in}

Thus in figure b, $\displaystyle {\int_a^bf}$ represents the shaded area with the area of the thick box subtracted from it, which is the same as the area of the region marked ``$+$" in figures c and d, with the area of the region marked ``$-$" subtracted from it.

The figure represents a geometric interpretation for a Riemann sum. In the figure

\psfig{file=ch8c.eps,width=5in}

\begin{displaymath}f(s_i)>0 \mbox{ for } i=1,2,3,\quad f(s_i)<0 \mbox{ for } i=4,5.\end{displaymath}


\begin{displaymath}\sum_{i=1}^3 f(s_i)(x_i-x_{i-1})\end{displaymath}

is the area of $\displaystyle {\bigcup_{i=1}^3 B\Big(x_{i-1},x_i\colon
0,f(s_i)\Big)}$ and

\begin{displaymath}\sum_{i=4}^5f(s_i)(x_i-x_{i-1})\end{displaymath}

is the negative of the area of

\begin{displaymath}\bigcup_{i=4}^5 B(x_{i-1},x_i:f(s_i),0).\end{displaymath}

In general you should think of $\displaystyle {\int_a^bf}$ as the difference $\alpha
(S^+)-\alpha (S^-)$ where

\begin{displaymath}S^+=\{(x,y)\colon a\leq x\leq b \mbox{ and } 0\leq y\leq f(x)\}\end{displaymath}

and

\begin{displaymath}S^-=\{(x,y)\colon a\leq x\leq b \mbox{ and } f(x)\leq y\leq 0\}.\end{displaymath}


8.6   Exercise. The graphs of two functions $f,g$ from $[0,2]$ to $\mbox{{\bf R}}$ are sketched below.
\psfig{file=ch8d.eps,width=5in}
Let

\begin{displaymath}F(x)=\Big( f(x)\Big)^2 \mbox{ for } 0\leq x\leq 2,\quad G(x)=\Big( g(x)\Big)^2
\mbox{ for } 0\leq x\leq 2.\end{displaymath}

Which is larger:
a)
$\displaystyle {\int_0^1f}$ or $\displaystyle {\int_0^1 F}$?
b)
$\displaystyle {\int_0^1 g}$ or $\displaystyle {\int_0^1 G}$?
c)
$\displaystyle {\int_0^1f}$ or $\displaystyle {\int_0^1 g}$?
d)
$\displaystyle {\int_0^{1/2}g}$ or $\displaystyle {\int_0^{1/2}G}$?
e)
$\displaystyle {\int_0^2 g}$ or $\displaystyle {\int_0^2 G}$?
Explain how you decided on your answers. Your explanations may be informal, but they should be convincing.

8.7   Exercise. Below is the graph of a function $g$. By looking at the graph of $g$ estimate the following integrals. (No explanation is necessary.)
\psfig{file=pieclin.eps,angle=-90,width=2.25in}
Graph of $g$

a)
$\displaystyle { \int_{1\over 4}^{3\over 4} g}$.
b)
$\displaystyle { \int_1^2 g}$.
c)
$\displaystyle { \int_0^{3\over 4} g}$.

8.8   Exercise. Sketch the graph of one function $f$ satisfying all four of the following conditions.
a)
$\displaystyle { \int_0^1 f = 1}$.
b)
$\displaystyle { \int_0^2 f = -1}$.
c)
$\displaystyle { \int_0^3 f = 0}$.
d)
$\displaystyle { \int_0^4 f = 1}$.



next up previous index
Next: 8.2 Properties of the Up: 8. Integrable Functions Previous: 8. Integrable Functions   Index
Ray Mayer 2007-09-07