Chapter 14 Review
Important Terms, Symbols, Concepts
 14.1 Area Between Curves
 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
 b
[ ( )

( )] a

Barnett/Ziegler/Byleen Business Calculus 11e 1

Chapter 14 Review
 14.1 Area Between Curves (continued)
 A graphical representation of the distribution of income among a population can be obtained by plotting data points ( x , y ) where x represents the cumulative percentage of families at or below a given income level, and y represents the cumulative percentage of total family income received.
Regresssion analysis can be used to find a function called a Lorenz curve that best fits the data.
Barnett/Ziegler/Byleen Business Calculus 11e 2

Chapter 14 Review
 14.1 Area Between Curves (continued)
 A single number, the Gini Index , measures income concentration:
Gini Index =
1
2 [
0
 x

( )]
 A Gini index of 0 indicates absolute equality - all families share equally in the income. A Gini index of 1 indicates absolute inequality - one family has all of the income and the rest have none.
Barnett/Ziegler/Byleen Business Calculus 11e 3

Chapter 14 Review
 14.2 Applications in Business and Economics
 Probability Density Functions.
If any real number x in an interval is a possible outcome of an experiment, then x is said to be a continuous random variable .
The probability distribution of a continuous random variable is described by a probability density function f that satisfies
 f ( x ) > 0 for all real x.


The area under the graph of f ( x ) over the interval
(-

,

) is exactly 1.
d
The probability that c < x < d is c
 f ( )
Barnett/Ziegler/Byleen Business Calculus 11e 4

Chapter 14 Review
 14.2 Applications (continued)
 Continuous Income Stream.
If the rate at which income is
 received - its rate of flow - is a continuous function f ( t ) of time, then the income is said to be a continuous income stream . The total income produced by a continuous income stream from t = a to t = b is b
 a f ( )
The future value of a continuous income stream that is invested at rate r , compounded continuously for 0 < t < T , is

T
FV

( )
(

) dt
0
Barnett/Ziegler/Byleen Business Calculus 11e 5

Chapter 14 Review
 14.2 Applications (continued)


Consumers’ and Producers’ Surplus.
If is a point on the graph of a price-demand equation p = D ( x ), then the consumers’ surplus p
CS
 x
0


D ( x )
 p
 dx
Similarly, for a point x p on the graph of a pricesupply equation p = S (x), the producers’ surplus at a
Barnett/Ziegler/Byleen Business Calculus 11e
PS
 x
0

 p

S ( x )
 dx
6

Chapter 14 Review
 14.3 Integration by Parts
 Some indefinite integrals, but not all, can be found by means of the integration by parts formula :
 udv
 uv
  vdu
 Select u and dv with the help of the guidelines in the section.
 14.4 Integration Using Tables
 A table of integrals is a list of integration formulas that can be used to find indefinite or definite integrals of frequently encountered functions. Such a list appears in
Table II of Appendix C.
Barnett/Ziegler/Byleen Business Calculus 11e 7