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\centerline{Mathematics 134 -- Calculus 2 With Fundamentals}
\centerline{Exam 3 -- Review Sheet}
\centerline{April 5, 2018}

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\noindent
{\it General Information}
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As announced in the course syllabus, the third midterm exam of the semester
will be given Friday, April 13 (not a bad omen, I hope!).  
The format will be similar to that of the other midterms.

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\begin{itemize}
\item You may use a calculator -- graphing calculators are OK, although I
doubt you'll find any use for the graphing features on this exam.
\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 5, 6 and the material
on improper integrals from last week), namely
the following material from sections 7.2, 7.3, 7.5, 7.6, 7.7 of Rogawski/Adams:
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\begin{enumerate}
\item Trigonometric integrals using the trig reduction formulas -- \emph{I will provide a copy
of the summary sheet on the reduction formulas for you to use}
\item Integration by trigonometric substitution
\item Integration by partial fractions
\item Improper integrals (integrals of functions with discontinuities and 
integrals over infinite intervals)
\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.
The methods of integration tested on the first two exams (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{in effect a cumulative exam on the material on methods of integration}.  
Especially there were things you had not mastered on the first two exams, 
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.  
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\noindent
There will be a review for the exam in class on Thursday, April 12.

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\noindent
{\it Review Problems from the text}
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\noindent
Chapter 7 review problems: 3 (except change the directions to say ``using the trigonometric reduction formulas''), 5, 6, 7, 8, 10 (a more specific direction: multiply the top and the bottom by $x$)
14, 15, 16, 19, 23, 27, 34, 42, 43, 75, 77, 82, 83
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\noindent
{\it Sample Exam Questions}
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\noindent
{\it Disclaimer: The actual exam questions may be organized differently.  
In particular, I may ask one question about the trigonometric substitution and/or
partial fractions methods the way I did on the quizzes instead of asking you to work out 
an integral from start to finish.  This list is also somewhat longer than the actual 
exam will be (to give you some more problems to practice on).}
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\noindent
I.  Compute each of the integrals below the appropriate method or
 combination of methods.  You must show all work
for full credit. 
\begin{enumerate}
\item[A)]  Use the trigonometric reduction formulas to find
$$\int \sin^3(x) \cos^4(x)\ dx$$
\item[B)]  Use the trigonometric reduction formulas to find 
$$\int \sec^5(x)\ dx$$
\item[C)]  $$\int \sqrt{25 - x^2}\ dx$$
\item[D)]  What changes in part C if the integral is 
$$\int \sqrt{25 + x^2}\ dx$$ 
Work this one out too, start to finish.  
\item[E)]  $$\int \frac{x^3 + 2x + 1}{x^2 + 6x - 7}\ dx$$
\item[F)]  $$\int \frac{1}{x^2(x^2 + 6x + 10)}\ dx$$
\end{enumerate}
II.  (Improper integrals)
\begin{enumerate}
\item[A)]  Why is the following integral an \emph{improper integral}.
Decide whether it converges by setting up and evaluating the appropriate
limit(s):
$$\int_1^3 \frac{1}{(x - 3)^{2/3}}\ dx$$
\item[B)]  Follow-up to part A:  For which exponents $\alpha > 0$ will the improper integral
$$\int_1^3 \frac{1}{(x - 3)^\alpha}\ dx$$
converge?
\item[C)]  Why is the following integral an improper integral?
Decide whether it converges by setting up and evaluating the 
appropriate limit(s):
$$\int_0^\infty x e^{-x}\ dx.$$
\end{enumerate}
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\noindent
III.  
\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 computation of the integral.
\item[(C)]  Our textbook's table of integrals gives this one as
$$\frac{-1}{a}\ln\left|\frac{a + \sqrt{a^2 - u^2}}{u}\right| + C$$
Show that the form you got in part B is equivalent to this.  
\end{enumerate}
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