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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Exam 2 -- Things To Know}
\centerline{March 15, 2002}
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\noindent
{\it General Information}
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As announced in the course syllabus, the second full-period
midterm exam of the semester will be given in class on Friday, 
March 22.  The format will be similar to that of the first midterm exam,
and the midterms
last semester.  The exam will cover the material since the 
first exam (Quizzes 4,5,6), including the following material
from Chapter 7, sections 2 and 3, and Chapter 8, sections 1,2,3.
{\it Note:  The material on numerical integration techniques
from Chapter 7, sections 5 and 6 that we used in Lab 2 
will not be covered on this exam, because we only used it via
the computer.}
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\item{1)}  Integrals by parts: Selecting appropriate
$u$ and $dv$, computing $du$ and $v$, using the parts formula
$\int u\ dv = uv - \int v\ du$, then finishing the integral on 
the right.  Recall that this might involve using parts again,
or another method such as substitution.
\item{2)}  Integrals by the table:  Recognizing the appropriate
entry, using reduction formulas repeatedly if necessary.
Be aware that a preliminary substitution might be necessary
to take an integral to one of the forms in the table.
\item{3)}  Areas and volumes by integration -- know how to 
set up an integral formula by slicing, approximating the area or
volume of the slice, summing, then taking a limit to get an integral.
You may need to use any one of the methods we know to evaluate
one of these.
\item{4)}  Important special case of 3 -- volumes of solids
of revolution.  Know how to distinguish between the cases of disk or 
ring (``washer'') cross-sections, and how to set up the integrals
in each case.
\item{5)}  Total mass and center of mass of an object of specified
shape and given density function.
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\noindent
There will be a review for the exam in class on Thursday, March 21.  
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\noindent
{\it Sample Exam}
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\noindent
I.  Compute each of the integrals below using basic rules, substitution,
or integration by parts.  You must show all work
for full credit.  Just quoting an entry from the integral
table will result in a deduction of 5 points per problem.
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\item{A)} $\int x^2 e^{-2x}\ dx$
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\item{B)} $\int x^3 \sin(x^2)\ dx$
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\item{C)} $\int x \arctan(x)\ dx$
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\noindent
II.  Compute each of the following integrals using an entry or 
entries from the table of integrals.  Say which entry or entries you 
are using.
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\item{A)} $\int \sqrt{x^2 - 9}\ dx$
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\item{B)} $\int \cos^4(6x)\ dx$
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III. {\it``Essay''} What are the ``rules of thumb''
we have used to select a good choice of $u$ and $dv$ in doing an 
integral by parts?
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IV.  Let $R$ be the region bounded by $y = x^{3/2} + 1$, $y = 0$,
$x = 0$ and $x = 4$.  
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\item{A)}  Sketch the region.
\item{B)}  Find the volume of the solid obtained if $R$ is rotated
about the $x$-axis.
\item{C)}  Find the volume of the solid obtained if $R$
is rotated about the line $y = -2$.
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\noindent
V.  A thin metal plate has the shape of the region in the plane
bounded by $y = \sin(x)$, $y = 0$, $x = \pi/4$, and $x = \pi/2$.  The 
density of the material of the plate at all points $x$ units from
the left-hand edge is $\rho(x) = x$ grams per unit area.
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\item{A)}  Compute the total mass of the plate.
\item{B)}  Set up (but do not evaluate) the integral to 
compute the $x$-coordinate of the center of mass.
\item{C)}  Will the $y$-coordinate of the 
center of mass of this plate be on the $x$-axis?  Why or why not?
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\noindent
Also see Problem Sets 4,5,6,7, the last three quizzes, and 
the Review Problems from the ends of Chapters 7,8 in
the text for other practice problems.
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