\documentclass[12pt]{article}

\usepackage{latexsym,amsmath,epsfig,graphicx}
\setlength{\textwidth}{6.5in} \setlength{\textheight}{9in} \oddsidemargin = -.1 cm
\evensidemargin = -.1 cm \topmargin=-2cm
\begin{document}

\centerline{Mathematics 132 -- Calculus for Physical and Life Sciences 2}
\centerline{Exam 2 -- Review Sheet}
\centerline{March 11, 2008}

\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
As announced in the course syllabus, the second midterm exam of the semester
will be given (``main seating'') at 6:00pm on Wednesday, March 26 (the first Wednesday
after Easter break).  The format will be similar to that of the first midterm
and the exams from last semester.

\vskip 10pt
\begin{itemize}
\item {\bf Bring a photo ID to the exam}.
\item The exam will be designed to take an hour but you will have an extra 30 
minutes to work and check your solutions.
\item You will be given a TI-30 scientific calculator for the exam which
does NOT have graphing capabilities so be prepared to answer questions
without your personal calculator.  (Note: Some of you may have one of these
calculators purchased for use in Chemistry courses here.  That is also OK.)
\item Use of cell phones, I-pods, and all other electronic devices \emph{is not
allowed} during the exam.  Please leave such devices in your room
or put them away in your backpack  (make sure cell phones are turned off).
\end{itemize}
\vskip 10pt
\noindent
{\it What will be covered}
\vskip 10pt
The exam will cover the material since the last exam (Problem Sets 5 and 6), namely
the following material from sections 5.9, 5.10, 6.1, 6.2, 6.3, and 6.4 of Stewart:
\vskip 10pt
\begin{enumerate}
\item Approximate integration using left- and right-hand Riemann sums, midpoint 
Riemann sums, trapezoidal and Simpson's rules.  Know how to compute each of these
with just a calculator (for a relatively small $n$).  Also know how to determine
whether your result is an under- or overestimate for the Riemann sums and the 
trapezoidal rules (and if it is possible to tell that).  You will want to review
the work we did in clas to prepare and your work on Lab 1 for much of this.
\item Improper integrals (know how to tell what makes an integral improper, how
to correctly set up the limit to determine if the integral converges, and determine
the value if the integral does converge).
\item Areas between curves
\item Volumes of solids with known cross-sections and solids of revolution
\item Arc length of curves
\item Average value of $f(x)$ on an interval $[a,b]$.
\end{enumerate}
\vskip 10pt
\noindent
\emph{Important Note:}  Most of the problems on this exam will require you 
to set up and compute an integral to find the quantity that is asked for.
In addition to knowing how to set up the required integral, \emph{any} of
the methods of integration tested on the first exam (i.e. basic rules, 
$u$-substitution, integration by parts, trigonometric substitution, partial 
fractions, or consultation of a table of integrals) might be required
to evaluate the integral.  In other words, this exam is \emph{really a 
cumulative exam on the first two-thirds of the semester}.  Especially 
if you did poorly on the first exam, you will need to begin your review
for this exam by going back and looking at the material from 
sections 5.1 through 5.8 in Stewart.
\vskip 10pt
\noindent
There will be a review for the exam in class on Tuesday, March 25.  \emph{Please
do not postpone preparing for this exam until after you return from 
Easter break -- you will not have enough time to get ready!}

\vskip 10pt
\noindent
{\it Review Problems}
\vskip 10pt
\noindent
Section 5.9/1, 3, 11, 13, 15
\vskip 10pt
\noindent
Section 5.10/1, 13, 15, 19 (use parts), 21, 23, 25, 29, 31, 55
\vskip 10pt
\noindent
Section 6.1/1, 3, 5, 7, 9, 11, 29 (place the origin at the center of 
the large circle, and note that the endpoints of the arc on the smaller
circle are at opposite ends of a diameter of that circle), 37
\vskip 10pt
\noindent
Section 6.2/1, 3, 5, 9, 21 (and evaluate), 29, 39 a (and evaluate)
\vskip 10pt
\noindent
Section 6.3/5, 7, 9 (look for an algebraic simplification to integrate), 
17, 19 (use the table for these)
\vskip 10pt
\noindent
Section 6.4/1, 3, 5, 13
\noindent
{\it Sample Exam Questions}
\bigskip
\noindent
This list is much longer than the actual exam will be (to give you some
idea of the range of different questions that might be asked).  
Unless otherwise directed, you may use any entry of the Table of 
Integrals from the text that applies.
\vskip 10pt
\noindent
I.
\begin{enumerate} 
\item[(A)] Use the midpoint rule, the trapezoidal rule, and Simpson's rule 
with $n = 4$ to estimate the value of the integral $\int_0^2 \sqrt{1 + x^4}\ dx$, 
rounding your answers to 6 decimal places.
\item[(B)] Which of your answers in part A are overestimates and which
are underestimates.  Explain how you can tell.
\item[(C)] Which of the estimates in part A would you expect to have the
smallest error.  Explain.  (Review Lab 1 to see the idea here.) 
\end{enumerate}
\vskip 10pt
\noindent
II.  For each of the following integrals, say why the integral is improper,
determine if the integral converges, and if so, find its value.
\vskip 10pt
\begin{enumerate}
\item[A)]  $\int_1^\infty \frac{1}{{\root 5 \of x}}\ dx$
\item[B)]  $\int_0^2 \frac{dx}{x^2 - 7x + 6}$
\item[C)]  $\int_0^\infty x e^{-3x}\ dx$
\item[D)]  For which values of $a$ is $\int_0^\infty e^{ax}\sin(x)\ dx$
convergent?  Evaluate the integral for those $a$. 
\end{enumerate}
\vskip 10pt
\noindent
III.
\begin{enumerate}  
\item[(A)]  Let $R$ be the region in the plane bounded by $y = 3 - x^2$ and the $x$-axis.
\begin{enumerate}
\item[(1)]  Sketch the region $R$.  
\item[(2)]  Find the area of $R$.
\item[(3)]  Find the volume of the solid generated by rotating $R$ about the
$x$-axis.
\end{enumerate}
\item[(B)]  Let $R$ be the region in the plane bounded by $y = 3x$ and $y = x^2$.  
\begin{enumerate}
\item[(1)]  Sketch the region $R$.
\item[(2)]  Find the area of $R$.
\item[(3)]  Find the volume of the solid generated by rotating $R$ about the
$x$-axis.
\item[(4)]  Find the volume of the solid generated by rotating $R$ about the
$y$-axis.
\end{enumerate}
\item[(C)]  Let $R$ be the region in the plane bounded by $y = \cos(\pi x)$, $y = 1/2$,
$x = -1/3$ and $x = 1/3$.
\begin{enumerate}
\item[(1)]  Sketch the region $R$.  
\item[(2)]  Find the area of $R$.
\item[(3)]  Find the volume of the solid generated by rotating $R$ about the
$x$-axis.
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
IV.  The height of a monument is 20m.  The horizontal cross-section of
the monument at $x$ meters from the top is an isosceles right triangle
with legs $x/4$ meters.  Find the volume of the monument.  
\vskip 10pt
\noindent
V. 
\begin{enumerate}
\item[(A)] Set up and evaluate the integral to compute the arclength of the 
curve $x = 3t^2$, $y = 2t^3$, $0 \le t \le 2$.
\item[(B)]  Set up and evaluate the integral to compute the arclength of the 
curve $y = \frac{1}{6}(x^2 + 4)^{3/2}$, $0 \le x \le 3$.  (Hint: the arclength
integral simplifies to a manageable form if you are careful with the algebra.)
\end{enumerate}
\vskip 10pt
\noindent
VI.  
\begin{enumerate}
\item[(A)] Find the average value of $f(x) = \sqrt{1 - x^2}$ on the interval
$[0,1/2]$.  (Use trigonometric substitution, not the table.)
\item[(B)] Find the average value of $f(x) = x \sqrt{1 + x^4}$ on the interval $[0,2]$. 
\end{enumerate}
\end{document}  
