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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Exam 2 -- Things To Know}
\centerline{March 17, 2006}
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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 24.  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 2,3,4,5,6, and Chapter 8, sections 1,2,3,7,8:
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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 partial fractions: Division 
to get degree of numerator less than the degree of the denominator,
factorization of the denominator, setting up partial fractions,
solving for coefficients, integration of resulting simpler
rational functions.
\item{3)}  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{4)}  Left-hand sum, Midpoint sum, Right-hand sum,
Trapezoidal rules for approximate numerical integration -- 
I might ask you to compute an approximate value with one or
more of these methods, but only for simple functions and
small $n$.  I might also ask ``qualitative'' questions
about which methods give over- or under-estimates in 
particular situations.  
\item{5)}  Applications of integrals:
\itemitem{a)}  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.  Important special case -- 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.
\itemitem{b)}  Total mass and center of mass of an object of specified
shape and given density function.
\itemitem{c)}  Arc length of curves
\itemitem{d)}  Probability density functions, cumulative distribution
functions, probabilities, means. 
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\noindent
There will be a review for the exam in class on Wednesday, March 22. 
Rosie will also be available for ``last minute'' questions
Thursday evening at the usual time. 
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\noindent
{\it Sample Exam Questions}
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\noindent
{\it Note:} The actual exam will only have 7 or 8 ``parts;'' this list
is somewhat longer than the actual exam will be.
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\noindent
I.  Compute each of the integrals below using integration by parts
or partial fractions: 
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\item{A)} $\int x^2 e^{-2x}\ dx$
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\item{B)} $\int {x + 1\over x^3 - 4x}\ dx$
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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. Which of the methods LEFT, RIGHT, MID, TRAP
is guaranteed to give an overestimate for $\int_0^4 f(x)\ dx$ for
all numbers $n$,  where the graph $y = f(x)$ is the following?
\vglue 2.0truein
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(There may be more than one!)
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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)}  Set up the integral to compute volume of the 
solid obtained if $R$ is rotated about the line $y = 10$.
\item{D)}  What is the length of the top boundary curve of the 
region $R$?
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\noindent
V.   A thin metal plate has the shape of the region in the plane
bounded by $y = \sin(x)$, $y = 0$, $x = 0$, and $x = \pi/2$.  The 
density of the material of the plate at $x$ 
is $\rho(x) = x + 1$ 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 formula to 
compute the $x$-coordinate of the center of mass.
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VI.  
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\item{A)}  For which value $c$ is $p(x)$ above a 
valid probability density function (pdf).  
\item{B)}  A random value is selected with distribution
described by $p(x)$.  What is the probability that
$1 < x < 2$?
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\noindent
Also see Problem Sets 4,5,6, the last three quizzes, and 
the Review Problems from the ends of Chapters 7,8 in
the text for other practice problems.
\bye
