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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{The definition of the Riemann Integral}

\begin{prop}
Rules  
\begin{enumerate}
\item \[
\int _{a}^{b}c\, dx=c(b-a).\]

\item \[
\int _{a}^{b}f(x)dx=-\int _{b}^{a}f(x)dx.\]
 Actually, this rule is a definition. We hadn't had a definition for the right-hand
side (at least when \( a<b \)); this helps for some later formulas. 
\item \[
\int _{a}^{a}f(x)dx=0.\]
 As before, this is really a definition. 
\item \[
\int _{a}^{b}cf(x)dx=c\int _{a}^{b}f(x)dx.\]
 
\item \[
\int _{a}^{b}f(x)\pm g(x)dx=\int _{a}^{b}f(x)dx\pm \int _{a}^{b}g(x)dx.\]
 
\item If \( f(x)\geq 0 \) for all \( x\in [a,b] \), \( a<b \), then \[
\int _{a}^{b}f(x)dx\geq 0.\]

\item If \( f(x)\geq g(x) \) for all \( x\in [a,b] \), \( a<b \), then \[
\int _{a}^{b}f(x)dx\geq \int _{a}^{b}g(x)dx.\]

\item For all \( a \), \( b \), and \( c \), \[
\int _{a}^{c}f(x)dx+\int _{c}^{b}f(x)dx=\int _{a}^{b}f(x)dx.\]

\item If \( m\leq f(x)\leq M \) for all \( x\in [a,b] \), \( a<b \), then \[
m(b-a)\leq \int _{a}^{b}f(x)dx\leq M(b-a).\]

\item \[
\left| \int _{a}^{b}f(x)dx\right| \leq \int _{a}^{b}\left| f(x)\right| dx.\]

\end{enumerate}
\end{prop}
\textbf{\emph{Proofs:}}

Rule \#1 is pretty obvious. Any Riemann sum for \( f(x)=c \) a constant will
have the same total, \( c \) times the length of the interval. Rules \#2 and
\#3 are, as mentioned, really definitions. They will be useful for some of the
later rules, though. Rule \#4 is also clear, since the \( c \) can be factored
out of any Riemann sum. 

Rule \#5 follows from the fact that, in any Riemann sum, \begin{eqnarray*}
\sum _{i=1}^{n}f\pm g(x_{i}^{*})\Delta x_{i} & = & \sum _{i=1}^{n}\left( f(x_{i}^{*})\pm g(x_{i}^{*})\right) \Delta x_{i}\\
 & = & \sum _{i=1}^{n}f(x_{i}^{*})\Delta x_{i}+\sum _{i=1}^{n}g(x_{i}^{*})\Delta x_{i},
\end{eqnarray*}
separating the Riemann sum into Riemann sums for \( f \) and \( g \) separately.
Because of this, the same separation holds for the integrals.

Rule \#6 is again obvious, since any Riemann sum would be positive, so must
be the integral. This rule is, however, often forgotten in the heat of evaluating
integrals using methods we'll deal with later on. This rule should be used as
a rough check on other work: if the function you are integrating is positive,
then the integral better be positive as well, or something went wrong.

Rule \#7 follows from \#6 and \#5. Since \( f(x)\geq g(x) \), then \( f(x)-g(x)\geq 0 \),
so \[
\int _{a}^{b}f(x)-g(x)\, dx\geq 0.\]
But\begin{eqnarray*}
0\leq \int _{a}^{b}f(x)-g(x)\, dx & = & \int _{a}^{b}f(x)\, dx-\int _{a}^{b}g(x)\, dx,
\end{eqnarray*}
so\[
\int _{a}^{b}f(x)\, dx\geq \int _{a}^{b}g(x)\, dx.\]


At the risk of repeating myself, again Rule \#8 is obvious, at least if \( a\leq b\leq c \),
since a Riemann sum for \[
\int _{a}^{b}f(x)dx\]


and for \[
\int _{b}^{c}f(x)dx,\]
when \emph{added together}, will give a Riemann sum for \[
\int _{a}^{c}f(x)dx.\]
 The result follows, in that case. If, on the other hand, the numbers \( a \),
\( b \), and \( c \) are not in increasing order, all you have to do is re-arrange
them in increasing order, making sure to change the sign of the integrals if
you switch order. It will all balance out.

Rule \#9 is just two applications of Rule \#7. 

Rule \#10 is a bit cleverer. Since\[
-|f(x)|\leq f(x)\leq +|f(x)|,\]
two more applications of Rule \#7 implies that \[
-\int _{a}^{b}|f(x)|dx\leq \int _{a}^{b}f(x)dx\leq +\int _{a}^{b}|f(x)|dx,\]
which becomes Rule \#10 when you turn that string of inequalities into an inquality
with absolute values (\( -A\leq B\leq A\iff |B|\leq A \)). This last rule is
often very useful theoretically.
\end{document}
