\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 134 -- Calculus 2 With Fundamentals}
\centerline{Exam 2 -- Review Sheet}
\centerline{March 2, 2018}

\vskip 10pt
\noindent
{\it General Information}
\vskip 10pt
As announced in the course syllabus, the second midterm exam of the semester
will be given Friday, March 16 (the first Friday after we return from 
Spring Break).  The format will be similar to that of the first midterm.

\vskip 10pt
\begin{itemize}
\item You will be permitted to use a calculator.  
\item Use of cell phones, I-pods, I-pads, tablets, and any 
other electronic device besides a calculator \emph{is not
allowed} during the exam.  Please leave such devices in your room
or put them away in your backpack (and 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 3, 4), namely
the following material from sections 6.1, 6.2, 6.3, 7.1 of Rogawski/Adams:
\vskip 10pt
\begin{enumerate}
\item Area between curves
\item Applications of integration and setting up integral formulas: volumes of solids with known 
cross-sections
\item Volumes of solids of revolution (disk and washer cross-sections, cylindrical shells as
an alternative method)
\item Integration by parts
\end{enumerate}
\vskip 10pt
\noindent
\emph{Important Note:}  Many 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,
the methods of integration tested on the first exam (i.e. basic rules, 
$u$-substitution) and the new methods on this
exam (integration by parts)
might be required to evaluate the integral.  In other words, this exam is 
\emph{in effect a cumulative exam on the material since the start of the semester}.  
Especially if there were things you had not mastered on the first exam, 
you will probably want to begin your review for this exam by going back and 
looking at the material from sections Chapter 5 in Rogawski/Adams.  
\vskip 10pt
\noindent
There will be a review for the exam in class on Thursday, March 15 and Tori will
be doing her regular Thursday session that evening for last-minute questions.

\vfill\eject
\noindent
{\it Review Problems}
\vskip 10pt
\noindent
Chapter 6 review problems: 1, 3, 5, 7, 9, 27, 29, 31, 33.  
\vskip 10pt
\noindent
Chapter 7 review problems: 3, 11, 13, 27, 31, 53, 57 
\vskip 10pt
\noindent
{\it Sample Exam Questions}
\vskip 10pt
\noindent
{\it Disclaimer: The actual exam questions may be organized differently and ask questions in 
different ways.  This list is also quite a bit longer than the actual exam will be (to give you some
idea of the range of different questions that might be asked).}
\vskip 10pt
\noindent
I.  Compute each of the integrals below using some combination of
basic rules, $u$-substitution, integration by parts.  You must show all work
for full credit. 
\begin{enumerate}
\item[A)]  $$\int \frac{e^{\tan^{-1}(x)}}{1 + x^2}\ dx$$
\item[B)]  $$\int x^2 \sin(2x) \ dx$$
\item[C)]  $$\int \sin^3(x) \cos(x)\ dx$$
\item[D)]  $$\int_0^{1/2} \sin^{-1}(x)\ dx$$
\item[E)]  Use integration by parts to show this reduction formula:
If $n$ is a positive integer, then
$$\int x^n e^{ax}\ dx = \frac{x^n e^{ax}}{a} - \frac{n}{a} \int x^{n-1} e^{ax}\ dx$$
\item[F)]  Apply the result from part (E) (repeatedly) to compute $\int x^4 e^{-2x}\ dx$.  
\item[G)]  $$\int x^8 \ln(x)\ dx$$
\end{enumerate}
%\vskip 10pt 
%\noindent
%II.  
%\begin{enumerate}
%\item[(A)]  Verify that $\int \csc\theta\ d\theta = \ln|\csc\theta - \cot\theta| + C$
%by differentiating.
%\item[(B)]
%Which trigonometric substitution would you apply
%to compute $\int \frac{1}{u \sqrt{a^2 - u^2}}\ du$?
%What trigonometric integral do you get after making the substitution?
%Complete the derivation of the integral.
%\item[(C)]  Our textbook's table of integrals gives this one as
%$$\frac{-1}{a}\ln\left|\frac{a + \sqrt{a^2 - u^2}}{u}\right|$$
%Show that the form you got in part B is equivalent to this.  
%\end{enumerate}
%\vskip 10pt
%\noindent
%II.
%\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
%III.  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
II.
\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.
\item[(4)]  Find the volume of the solid generated by rotating $R$ about the line
$y = 5$.
\item[(5)] Find the volume of the solid generated by rotating $R$ about the line
$y = -2$.
\item[(6)] Find the volume of the solid generated by rotating $R$ about the line
$x = -4$.  (Note: This can be done ``by slices'' or by ``shells.''  Try it 
both ways and check that you get the same result.    
\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.
\item[(4)]  Find the volume of the solid generated by rotating $R$ about the line $y = -1$.  
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
III.  (``Thought question'') 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.  Is the given information enough to find the volume of the monument?
(Hint: Look up {\it Cavalieri's Principle}.)  
%\vskip 10pt
%\noindent
%V.  A solid paperweight has a circular base of radius 4 cm.  The cross-sections of the paperweight
%by planes perpendicular to one diameter of the base are equilateral triangles.  Find the volume
%of the paperweight.  
%\vskip 10pt
%\noindent
%VI.  A \emph{45 rpm single} record was a vinyl disk 7 inches in diameter, with a large central hole 1 inch in 
%diameter.  A ring-shaped paper label 1 inch wide was usually glued to the vinyl surface outside the hole, but
%inside the grooves where the sound was recorded.  The vinyl making up the disk, plus the paper of the 
%label, had mass density .1 ounce per square inch at all points outside the hole in the central region.
%But the density then decreased linearly from .1 to .07 ounces per square inch at the outer edge.  
%The density was the same at all points in the region the same distance away from the center of the hole.  
%Set up an integral or integrals to compute the total mass of a 45 rpm single record (including the label)
%and compute the total mass.  
%\vskip 10pt

\end{document}  
