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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Exam 2 -- Things To Know}
\centerline{March 19, 2001}
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{\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 23.  The format will be similar to that of the midterm exams
last semester and Exam 1 this time.  The exam will cover the material
since the last 3xam (Quizzes 4, 5, 6), including the following material from 
Chapter 7, section 7, Chapter 8 sections 1,2, and the "Focus on
Modeling" section at the end of Chapter 8:
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\item{1)}  Improper integrals (integrals over infinite intervals,
and integrals of functions with discontinuities in the interval 
$[a,b]$) -- be able to identify which kind of improper integral
you have, and set up the appropriate limits to evaluate them/determine
whether they converge or not
\item{2)}  Integrals for computing volumes of solids with known 
cross-sections:  $V = \int_a^b A(x)\ dx$, where $A(x)$ is the 
cross-section area function.
\item{3)}  Volumes of solids of revolution -- Note:  This is a special
case of 2.
\item{4)}  Integrals for computing total mass/center of mass of
wires and plates of given shapes, given the mass density function
\item{5)}  Density functions of distributions, cumulative 
distribution function, median, mean.
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A copy of the table of integrals from our text will be provided.
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There will be a review for the exam in class on Thursday, March 22.  
I will also set up an evening session with Jodi Marani on Thursday for 
last minute questions.
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{\it Sample Exam}
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I.  Two of the following three integrals are improper integrals.
Say which are the improper ones, and evaluate those only.
\item{A)}  $\int_0^\infty {1\over 9x^2 + 1}\ dx$
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\item{B)}  $\int_1^2 {1\over x - 7}\ dx$
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\item{C)}  $\int_2^4 {1\over (x-2)^{3/4}}\ dx$
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II.  Let $R$ be the region bounded by $y = x^2$, $y = 0$,
$x = 0$ and $x = 1$.  
\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 $y = 4$.
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III.  A hill with a circular base has height $z = 100 e^{-r^2/10}$ 
meters above the surrounding ground level at all points $r$ meters 
measured horizontally from the center of the base.
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\item{A)}  How could you subdivide the hill so that the height
is roughly constant on each portion of the hill?   
\item{B)}  Use your idea from A to set up an integral giving
the total volume of the hill above the surrounding 
ground level.  Evaluate to find the volume.
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IV.  Let $x$ be the radius in millimeters of raindrops 
during a storm.  Suppose the density function $p(x)$ 
for $x$ has $p(1) = .4$, is linear
over the range $1 \le x \le 6$ and is zero everywhere else.
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\item{A)}  Find a formula for $p(x)$.
\item{B)} Plot the cumulative distribution function for the radius of 
the raindrops, $P(x)$.  
\item{C)}  What is the mean radius of the raindrops?

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Also see Problem Sets 4,5, the last three quizzes, and 
the Review Problems from the ends of Chapters 7,8 in
the text for other practice problems.
\bye
