\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{multicol}


\title{Math 1a.  Lecture 10 \\
Applying the Definite Integral: Slice and Conquer}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to use Riemann Sums and the definite integral to calculate ``mass'' when density varies.


\end{itemize} 



\section{Worksheet Examples}

\begin{enumerate}

\item
A city is in the shape of a rectangle 4 miles wide and 6 miles long.  A river runs through the middle of the city, parallel to the 6-mile long sides.  People prefer to live nearer the water, so the density of people is given by $\rho(x) = 10000 - 800x$ people per square mile, where $x$ is the distance from the river.
\begin{enumerate}

\item
Show in a sketch how to slice up the region.

\item
What is the area of the $i$th slice?

\item
What is the approximate population of the $i$th slice?

\item
Write a Riemann sum to estimate the total population of the city.

\item
Calculate the population of the city using a definite integral.

\end{enumerate}

\item
A village is in the shape of a circle of radius 3 kilometers.  A well, located in the center of the city, is the only source of water.  People prefer to live nearer the well, so the density of people is given by $\rho(x) = 800 - 100x$ people per square mile, where $x$ is the distance from the well.
\begin{enumerate}

\item
Show in a sketch how to slice up the region.

\item
What is the area of the $i$th slice?

\item
What is the approximate population of the $i$th slice?

\item
Write a Riemann sum to estimate the total population of the city.

\item
Calculate the population of the city using a definite integral.

\end{enumerate}


\item
A beam of light is shining onto a screen creating a disk of radius 50 cm.  The intensity of the light is brightest at the center and diminishes away from the center.  If the intensity of the light at a distance $r$ from the center of the bean is given by 
$$
f(r) = \frac{150}{20 + r^2}
$$
watts per cm$^2$, find the total wattage of the beam's image on the screen.

\item
Suppose the density of a mass in a gaseous planet is given by the function
$$
\rho(r) = \frac{40000}{1 + 0.0001r^3}
$$
kilograms per cubic kilometer, where $r$ is the distance in kilometers from the center of the planet.  Find the total mass of the planet if its radius is 8000 km.



\end{enumerate} 


\section*{References}

\begin{itemize}

\item
\S 27.1 in 
Robin Gottlieb.
{\it Calculus: An Integrated Approach to Functions and their Rates of Change}.
Addison Wesley, Boston, 2002.
ISBN 0--201--70929--5.


\end{itemize}



\section*{Notes}

\subsection*{March 3, 2006}



\end{document}

