Chapter 7
Additional
Integration
Topics
Section 1
Area Between Curves
Learning Objectives for Section 7.1
Area Between Two Curves
The student will be able to
1. Determine the area between a
curve and the x axis ,
2. Determine the area between two
curves, and
3. Solve an application involving
income distribution.
2
Negative Values
for the Definite Integral
If f (x) is positive for some values of x on [a, b] and negative
for others, then the definite integral

b
a
f ( x ) dx
is the cumulative sum of the signed
areas between the graph of f (x) and
the x axis. Areas above the x axis are
positive, and areas below are
negative. In this section we want find
the (unsigned) actual area between a
curve and the x axis or between two
curves. This is never negative!
y = f (x)
B
a
A

b
a
b
f ( x) dx   A  B
3
Negative Values
The (unsigned) area between the
graph of a negative function and the
x axis is equal to the definite
integral of the negative of the
function.
A

B
b
a

[  f ( x) ] dx
c
b
y = f (x)
B
a
A
b
c
f ( x) dx
4
Area Between Curve and x axis
The area between f (x) and the x axis can be found as follows:
For f (x) > 0 over [a, b] (i.e. f (x) is above the x axis):
Area =

b
a
f ( x) dx
For f (x) < 0 over [a, b] (i.e. f (x) is below the x axis):
Area =

b
a
[  f ( x) ] dx
5
Example
Find the area bounded by
y = 4 – x2 ;
y = 0,
1 ≤ x ≤ 4.
A1
A2
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Example
Find the area bounded by
y = 4 – x2 ;
y = 0,
1 ≤ x ≤ 4.
A1
A2
Area = A1 – A2


2
0
f ( x) dx 

4
2
f ( x) dx
x3 2 
x3  4
 4 x  |   4 x   |
3 0 
3 2
 8 – 8/3 – (16 – 64/3 – 8  8/3)  16
7
Area Between Two Curves
Theorem 1. If 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
 a  f ( x)  g ( x) dx
b
f (x)
A
a
b g (x)
8
Example
Find the area bounded by
y = x2 – 1 and
y = 3.
9
Example
Find the area bounded by
y = x2 – 1 and
y = 3.
Note the two curves intersect at –2
and 2, and y = 3 is the larger
function on –2 ≤ x ≤ 2.
A
2


2

3

x
1  dx
 2 
x3
32
2
 3x –  x 2 
 10.67
3
3
10
Computing Areas Using a
Numerical Integration Routine
Suppose we want to find the
area bounded by
f ( x)  e
 x2
g ( x)  x 2  1
11
Computing Areas Using a
Numerical Integration Routine
Suppose we want to find the
area bounded by
f ( x)  e
 x2
g ( x)  x 2  1
First, we use a graphing calculator to graph the functions f and g
and find their intersection points as in the figure. We see that the
graph of f is bell shaped and the graph of g is a parabola, and
that f (x) > g(x) on the interval [–1.131, 1.131].
12
Computing Areas Using a
Numerical Integration Routine
Then we compute the area
by a numerical integration
routine on a calculator.
13
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.
14
Application
(continued)
For example, data point
(approximately) (0.4, 0.09) in the
table indicates that the bottom 40%
of families receive only 9% of the
total income for all families.
Curve of
absolute
equality y = x
Lorenz
Curve
(0.4, 0.09)
Data point (.6, .26) indicates that
the bottom 60% of families receive
only 26% of the total income, etc.
15
Application
(continued)
If we have absolute inequality of income (one family has all
the income, the rest have none), the Lorenz curve is y(x) = 0
for x < 1, y(1) = 1.
The maximum possible area between the Lorenz curve and the
line y = x is ½, which occurs in the case of absolute inequality.
That is, the area between the Lorenz curve and the line
y = x is between 0 and ½.
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.
16
Application
(continued)
Gini Index of Income Concentration
If y = f (x) is the Lorenz curve, then
Index of income concentration = 2
 x 
1
0
f ( x) dx
A measure of 0 indicates absolute equality. A measure
of 1 indicates absolute inequality.
17
Example
A country is planning changes in tax structure in order to
provide a more equitable distribution of income. The two
Lorenz curves are:
f (x) = x2.3 currently, and
g(x) = 0.4x + 0.6x2 proposed.
Will the proposed changes work?
18
Example
(continue)
Currently: Gini Index of income concentration =
2
3.3

 1
x
x
2.3
|  0.3939
2  [ x  x ] dx  2 

0
3.3  0
 2
1
Future: Gini Index of income concentration =
2

1
0
[ x  (0.4 x  0.6 x 2 ] dx
 0.6 x 2 0.6 x 3  1
|  0.20
 2 

3  0
 2
The Gini index is decreasing, so the future distribution will be
more equitable.
19
Summary
■ We reviewed the definite integral as the area between a
curve and the x axis.
■ We learned how to calculate area when the curve was
below the x axis.
■ We learned how to calculate the area between two
curves.
■ We learned about the Lorenz curve and the Gini index
of income concentration.
Barnett/Ziegler/Byleen Business Calculus 12e
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