5128_Ch07_pp378-433

422 Chapter 7 Applications of Definite Integrals
SOLUTION
(a) At the bottom of the tank, the molasses exerted a constant pressure of
90 y k
45
Figure 7.45 The 1-ft band at the bottom
of the tank wall can be partitioned into thin
strips on which the pressure is approximately
constant. (Example 4)
p wh ( 100 lb
ft3) (90 ft) 9000 lb
ft2.
Since the area of the base was p45 2 , the total force on the base was
(
lb
9000 f t2) (2025 p ft2 ) 57,225,526 lb.
(b) We partition the band from depth 89 ft to depth 90 ft into narrower bands of width
Δy and choose a depth y k in each one. The pressure at this depth y k is p wh 100 y k
lbft 2 (Figure 7.45). The force against each narrow band is approximately
pressure area 100y k 90p Δy 9000p y k Δy lb.
Adding the forces against all the bands in the partition and passing to the limit as the
norms go to zero, we arrive at
F 90
89
9000pydy 9000p 90
ydy 2,530,553 lb
89
for the force against the bottom foot of tank wall. Now try Exercise 25.
1
12
y
2
Area
= 1
4
5
12
Figure 7.46 The probability that the
clock has stopped between 2:00 and 5:00
can be represented as an area of 14.
The rectangle over the entire interval has
area 1.
x
Normal Probabilities
Suppose you find an old clock in the attic. What is the probability that it has stopped
somewhere between 2:00 and 5:00?
If you imagine time being measured continuously over a 12-hour interval, it is easy to
conclude that the answer is 14 (since the interval from 2:00 to 5:00 contains one-fourth of
the time), and that is correct. Mathematically, however, the situation is not quite that clear
because both the 12-hour interval and the 3-hour interval contain an infinite number of
times. In what sense does the ratio of one infinity to another infinity equal 14?
The easiest way to resolve that question is to look at area. We represent the total probability
of the 12-hour interval as a rectangle of area 1 sitting above the interval (Figure 7.46).
Not only does it make perfect sense to say that the rectangle over the time interval 2, 5
has an area that is one-fourth the area of the total rectangle, the area actually equals 14,
since the total rectangle has area 1. That is why mathematicians represent probabilities as
areas, and that is where definite integrals enter the picture.
Improper Integrals
More information about improper
integrals like
fx dx can be found in
Section 8.3. (You will not need that
information here.)
DEFINITION Probability Density Function (pdf)
A probability density function is a function f x with domain all reals such that
f x 0 for all x and f x dx 1.
Then the probability associated with an interval a, b is
b
f x dx.
a
Probabilities of events, such as the clock stopping between 2:00 and 5:00, are integrals
of an appropriate pdf.