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\title{Math 1a.  Lecture 11 \\
Arc Length and Average Value of a Function}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to compute arc length.

\item
To understand and be able to compute the average value of a function.

\end{itemize} 



\section{Arc Length}

We can approximate the length of a curve $C$ be calculating the length of inscribed polygonal arc, $C_n$.  Let $C$ be given by the function $y = f(x)$ on the interval $[a, b]$ and consider a subdivision
$$
a = x_0 < x_1 < x_2 < \cdots < x_{n-1} < x_n = b.
$$
The length of the $i$th line segment in our polygonal arc is
$$
\sqrt{ [f(x_i) - f(x_{i-1})]^2 + (x_i - x_{i-1})^2}.
$$
Thus, the length of the entire polygonal arc is
\begin{eqnarray*}
|C_n|
& = &
\sum_{i = 1}^n \sqrt{ [f(x_i) - f(x_{i-1})]^2 + (x_i - x_{i-1})^2} \\
& = &
\sum_{i = 1}^n \sqrt{ (\Delta y_i)^2 + (\Delta x_i)^2} \\
& = &
\sum_{i = 1}^n \left(\sqrt{ \left( \frac{\Delta y_i}{\Delta x_i} \right)^2 + 1} \right) \cdot \Delta x_i
\end{eqnarray*}
Therefore, the length of the arc can be found by letting $n \to \infty$
$$
|C| = \int_a^b \sqrt{[f'(x)]^2 + 1} \, dx.
$$

\section{Example}

The length of $y = \sin x$ from $x = 0$ to $x = \pi$ is
$$
\int_0^{\pi} \sqrt{\cos^2 x + 1} \, dx \approx 3.8202.
$$

Most integrals coming from arc length problems cannot be evaluated in terms of elementary functions.

\section{Finding the Average Temperature}

\begin{enumerate}

\item
How would you find the average temperature in Boston over a 24-hour period?

\item
How would you calculate the average temperature in Boston given the following data?

\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{| r | l |} \hline
Time & Temperature \\ \hline
2 PM & 68 $^\circ$F \\
6 PM & 68 \\
10 PM & 58 \\
2 AM & 55 \\
6 AM & 54 \\
10 AM & 65 \\
2 PM & 74 \\ \hline
\end{tabular}
\end{center}

\item
How would you calculate the average temperature in Boston given that the temperature is given by
$$
f(t) = \frac{1}{9} t^2 - \frac{29}{12} t + 68,
$$
where $t$ is given in hours since 2 PM and $f(t)$ is given in degrees Fahrenheit?
\begin{enumerate}

\item
Partition the interval $[0,24]$ into $n$ equal subintervals, so that the length of each subinterval is $$
\Delta t = \frac{24}{n}$$ and $$t_i = 0 + i \Delta t.
$$

\item
Then the average of the temperatures taken at times $t_1$, $t_2$, \ldots, $t_n$ is
\begin{align}
\frac{f(t_1) + f(t_2) + \cdots + f(t_n)}{n}
&=\frac{f(t_1) + f(t_2) + \cdots + f(t_n)}{24/\Delta t} \notag\\
&= \frac{\left ( f(t_1) + f(t_2) + \cdots + f(t_n) \right ) \Delta t}{24} \notag\\
&= \frac{ f(t_1) \Delta t + f(t_2) \Delta t + \cdots + f(t_n) \Delta t}{24} \notag\\
&= \frac{1}{24} \sum_{i=1}^n f(t_i) \Delta t. \notag
\end{align}

\item
As $n \to \infty$, the average temperature converges to 
$$
\frac{1}{24} \int_0^{24} f(t)\,dt.
$$
	
\item In this case,
$$
\frac{1}{24} \int_0^{24} f(t)\,dt = \frac{181}{3} = 60 \frac{1}{3}.
$$
	
\end{enumerate} 

\end{enumerate}

\section{Average Value of a Function}

Define the {\em average value of a function} $f$ on the interval $[a,b]$ to be 
$$
\frac{1}{b-a} \int_a^b f(x)\,dx.
$$
Note that if we let $c$ be the average value of $f$ on the interval $[a,b]$, then 
$$
c (b-a) = \int_a^b f(x)\,dx.
$$ 
This means that the rectangle with base $[a,b]$ and height $c$ has the same area as the region under the graph of $f$ from $a$ to $b$.

\section{Example}


The graph of some function $f$ is given below.  List the following values from {\em smallest\/} to {\em largest}.   


\vspace{.2in}
\centerline {
\includegraphics[width=3in]{Fig_11_1}
}
\vspace{.2in}


\begin{enumerate}

\item
$f'(1)$

\item
The average value of $f(x)$ for $0 \leq x \leq a$.

\item
The average value of the rate of change in $f(x)$ for $0 \leq x \leq a$.

\item
$\displaystyle \int_0^a f(x) dx$.

\end{enumerate}


	
\section*{References}

\begin{itemize}

\item
\S 6.3--6.4 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{February 24, 2006}



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