496
CHAPTER 5
.
.
Applications of the Definite Integral
566
y
b
a
x
a
±
w
b
±
w
49.
For tennis rackets, a large second moment (see exercises 47
and 48) means less twisting of the racket on offcenter shots.
Compare the second moment of a wooden racket (
a
=
9
,
b
=
12
,w
=
0
.
5), a midsize racket (
a
=
10
,
b
=
13
=
0
.
5)
and an oversized racket (
a
=
11
,
b
=
14
=
0
.
5).
50.
Let
M
be the second moment found in exercise 48. Show that
dM
da
>
0 and conclude that larger rackets have larger second
moments. Also, show that
d
w
>
0 and interpret this result.
EXPLORATORY EXERCISES
1.
As equipment has improved, heights cleared in the pole vault
have increased. A crude estimate of the maximum pole vault
possible can be derived from conservation of energy princi
ples. Assume that the maximum speed a polevaulter could run
carrying a long pole is 25 mph. Convert this speed to ft/s. The
kinetic energy of this vaulter would be
1
2
m
v
2
. (Leave
m
as an
unknown for the time being.) This initial kinetic energy would
equal the potential energy at the top of the vault minus what
ever energy is absorbed by the pole (which we will ignore).
Set the potential energy, 32
mh
,
equal to the kinetic energy and
solve for
h
. This represents the maximum amount the vaulter’s
center of mass could be raised. Add 3 feet for the height of
the vaulter’s center of mass and you have an estimate of the
maximum vault possible. Compare this to Sergei Bubka’s 1994
world record vault of 20
±
1
3
4
±±
.
2.
An object will remain on a table as long as the center of mass
of the object lies over the table. For example, a board of length
1 will balance if half the board hangs over the edge of the table.
Show that two homogeneous boards of length 1 will balance
if
1
4
of the ﬁrst board hangs over the edge of the table and
1
2
of
the second board hangs over the edge of the ﬁrst board. Show
that three boards of length 1 will balance if
1
6
of the ﬁrst board
hangs over the edge of the table,
1
4
of the second board hangs
over the edge of the ﬁrst board and
1
2
of the third board hangs
over the edge of the second board. Generalize this to a pro
cedure for balancing
n
boards. How many boards are needed
so that the last board hangs completely over the edge of the
table?
L
2
L
4
. . .
5.7
PROBABILITY
The mathematical ﬁelds of probability and statistics focus on the analysis of random pro
cesses. In this section, we give a brief introduction to the use of calculus in probability
theory. It may surprise you to learn that calculus provides insight into random processes,
but this is in fact, a very important application of integration.
We begin with a simple example involving cointossing. Suppose that you toss two
coins, each of which has a 50% chance of coming up heads. Because of the randomness
involved, you cannot calculate exactly how many heads you will get on a given number of
tosses. But you
can
calculate the
likelihood
of each of the possible outcomes. If we denote
heads by H and tails by T, then the four possible outcomes from tossing two coins are HH,
HT, TH and TT. Each of these four outcomes is equally likely, so we can say that each has
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 Spring '10
 atsumi
 Probability distribution, Probability theory, probability density function, World Series, Blaise Pascal

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