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\centerline{Mathematics 136 -- Calculus 2}
\centerline{Exam 2 -- Review Sheet}
\centerline{March 21, 2014}

\vskip 10pt
\noindent
{\it General Information}
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As announced in the course syllabus, the second midterm exam of the semester
will be given Friday, March 28.  The format will be similar to that of the first midterm
and the exams from last semester.

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\begin{itemize}
\item You will not be given a calculator for the exam.  If you 
want to use one, you will be permitted to bring a TI-30 or similar
{\it non-graphing} 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, I-pads, tablets, and any 
other electronic device besides a basic 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}
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\noindent
{\it What will be covered}
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The exam will cover the material since the last exam (Problem Sets 4, 5 and 6), namely
the following material from sections 5.7, 5.8, 5.9, 5.10, 6.1, 6.2 of Stewart:
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\begin{enumerate}
\item Integration by trigonometric substitution and partial fractions
\item Use of the table of integrals (a copy of relevant portions will be provided
for your use on the exam)
\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
\end{enumerate}
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\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, \emph{any} of
the methods of integration tested on the first exam (i.e. basic rules, 
$u$-substitution, integration by parts) and the new methods on this
exam 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 there were things you had not mastered 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 5.6 in Stewart.
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\noindent
There will be a review for the exam in class on Wednesday, March 26.

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\noindent
{\it Review Problems}
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\noindent
Section 5.7/11, 13, 14, 15, 17, 19, 21, 23, 25, 27, 31
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\noindent
Section 5.8/1 - 21 odds
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\noindent
Section 5.9/1, 3, 11, 13, 15
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\noindent
Section 5.10/1, 13, 15, 19, 21 (do a $u$-substition first), 23, 25, 29, 31, 55
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\noindent
Section 6.1/1, 3, 5, 7, 9, 11, 29, 37
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\noindent
Section 6.2/1, 3, 5, 9, 27, 29, 31, 33, 39
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\noindent
{\it Sample Exam Questions}
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\noindent
This list is somewhat 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.
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\noindent
I.
\begin{enumerate}
\item[(A)]  Do you need partial fractions to 
compute 
$$\int {t^2 + 1\over t^3 + 3t + 3}\ dt?$$
Explain, and if possible compute the integral with a simpler method.
\item[(B)]  Apply partial fraction decomposition to 
compute $$\int {1\over x(x - 1)(x + 2)}\ dx$$
\item[(C)]  Which trigonometric substitution would you apply
to compute $\int {1\over u \sqrt{a^2 - u^2}}\ du$?
What trigonometric integral do you get after making the substitution?
Complete the derivation of the integral.
\end{enumerate}
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}
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\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.
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\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}
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\noindent
IV.
\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}
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\noindent
V.  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.  

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