5. Additional Integration Topics
5-1 Area Between Curves
Area Between Two Curves
Theorem 1
If f and g are continuous and f  x   g  x  over
the intervala, b, then the area bounded by
y  f  x  and y  g  x  for a  x  b is given by
a  f x   g x dx .
b
Find the area bounded by f  x   6 x  x 2 and
y  0 for 1  x  4 .
Find the area between the graph of f  x   x 2  2 x
and the x-axis over 1,2; over  1,1.
Find the area between the graphs of
1
f  x   x  3 and g  x    x 2  1 over  2,1.
2
Find the area between the graphs of
f  x   5  x 2 and g  x   2  2 x .
Find the area between the graphs of
f  x   x 2  x and g  x   2 x for  2  x  3.
Find the area to three decimal places bounded
 x2
by f  x   e and g  x   x 2  1.
Application: Income Distribution
The U.S. Bureau of the Census compiles data on
distribution of income among families. This data
can be fitted to a curve using regression analysis.
This curve is called a Lorenz curve.
The variable x represents the cumulative
percentage of families at or below the given
income level. The variable y represents the
cumulative percentage of total family income
received. If we have absolute equality of
income (every family has the same income), the
Lorenz curve is y  x .
Curve of
absolute
equality
yx
Lorenz
Curve
(0.4, 0.09)
Corrado Gini (1884-1965)
The Gini Index is two times the area between
the Lorenz curve and the line y = x. It can take
on values between 0 and 1.
Gini Index of Income Concentration
If y  f  x  of a Lorenz curve, then
Gini index = 2 x  f  x dx
1
0
The Lorenz curve in a certain country in 2010 is
f  x   x 2.6 . Economists predict that it will be
f  x   x1.8 by 2025. Find the Gini index for
each and interpret.
5-2 Applications in Business and Economics
Probability Density Functions
A probability density function must satisfy:
1. f  x   0 for all real x.
2. The area under the graph of f  x  is 1.
3. If c, d  is a subinterval then
Probability c  x  d    f  x dx
d
c
Suppose the length of phone calls is a
continuous random variable with probability
 1 e t 4 , t  0
density function f t    4
.
0, t  0
Determine probability of a call between 2 and 3
minutes.
Find b to two places so that the probability of a
call selected at random lasting between 2 and b
minutes is .5
Normal probability density function
2
1
 x    / 2 2
f x 
e
 2
d  x   2 / 2 2
1
e
dx
Probability c  x  d  

c
 2
Continuous Income Stream
Suppose you have a trust that pays $2000/yr.
How much income will you have by the 10th year?
10
0
2000dt
The rate of change of the income produced by a
vending machine is f t   5,000e0.04t for the
first t years of operation. Find the total income
produced during the first 5 years.
Total Income for a Continuous Income Stream:
If f (t) is the rate of flow of a continuous income
stream, the total income produced during the
time period from t  a to t  b is
a f t dt
b
Future Value of a Continuous Income Stream
What is the future value of $10,000 at 12%
compounded continuously for 5 years?
A  Pe rt
If f t  is the rate of flow of a continuous
income stream, 0  t  T , and if the income is
continuously invested at a rate r, compounded
continuously, then the future value FV at the
end of T years is given by
FV   f t e
T
0
r T t 
dt
rT T
e
0

f t e rt dt
Suppose you have a trust that pays $2000/yr,
which is immediately invested at 8%. How
much income will you have by the 10th year?
FV
rT T
e
0

f t e rt dt
The rate of change of the income produced by a
vending machine is f t   5,000e0.04t for the
first t years of operation. Find the future value
of this income stream at 12% compounded
continuously for 5 years, and the total interest
earned.
Consumers’ and Producers’ Surplus
Let p  D x  be a price-demand equation for a
product (x produced at $p/unit)
Consumers’ Surplus
If x, p  is a point on the graph of the pricedemand equation P  D x , the consumers’
surplus CS at a price level of p is
CS   D x   p dx
x
0
The consumers’ surplus represents the total
savings to consumers who are willing to pay
more than p but are still able to buy the product
at p .
Find the consumers’ surplus at a price level of
$8 for the price-demand equation
p  D x   20  0.05x .
Producers’ Surplus
If x, p  is a point on the graph of the pricesupply equation P  S  x , the producers’
surplus PS at a price level of p is
PS    p  S  x dx
x
0
The producers’ surplus represents the total gain
to producers who are willing to supply units at a
lower price than p but are still able to supply
units at p .
Find the producers’ surplus at a price level of
$20 for the price-supply equation
p  S  x   2  0.0002 x 2
Equilibrium price
Find the equilibrium price, then find the
consumers’ and producers’ surplus at the
equilibrium price level if
p  D x   20  0.05x and p  S  x   2  0.0002 x 2
y2  y1
2 1 
 2.71828  r  
x2  x1
 y
Force 

x 

 Acceleration 
2
1