%% Math 131, Fall, 2003
\documentclass[11pt]{article}
\usepackage{latexsym,amsmath,epic,eepic,pstcol}

\usepackage{math131}

\pagestyle{empty}
\newcommand{\ddx}[1]{\ds\frac{d}{d#1}}

\begin{document}

\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Professor Hwang} \\
Worksheet on Riemann Sums
\end{center}

\be

\item In the figure below, sketch the inscribed and circumscribed
  rectangles for 4~intervals. Then use your calculator to find upper
  and lower bounds for $\ds\int_1^2\frac{1}{t}\,dt$.

  \begin{center}
  \input{handout09a.eepic}
  \end{center}

  \bi
  \item By how much do your bounds differ? 
  \item What is your best guess of the actual value of the integral?
  \item The actual value of the integral is~$\ln 2\simeq0.69315$. How
  far off is your best guess?
  \item If 12~intervals are used instead of~4, by how much do the
  upper and lower bounds differ?
  \ei

\item Let $f(t)=4t^3+1$, and suppose we want to approximate
  $\ds\int_{-1}^1 f(t)\,dt$. (Note~$f$ is increasing.)

  \bd

  \item{(a)} If we use~100 intervals of equal length, by how much will
    the left-hand and right-hand estimates differ? (Do not compute the
    actual estimates!)
  
  \vfil

  \item{(b)} How many intervals would guarantee an accuracy of
    4~decimals? (In other words, how many intervals are required to make
    the difference between the upper and lower estimates smaller
    than~$10^{-4}$?) 6~decimals? Your answers should convince you that
    rectangles are not a practical way of estimating definite integrals.

  \vfil

  \ed

\ee

\eject

\end{document}
