6.5 Applications of the Definite Integral

In this section, we will introduce applications of the definite integral.

• Average Value of a Function• Consumer’s Surplus• Future Value of an Income Stream

The Average Value of a Function

Let f(x) be a continuous function on the interval bxa

The definite integral may be used to define the average value of f(x) on this interval.

The average value of a continuous function f(x) over the interval bxa is defined as the quantity

b

adxxf

ab)(

1

Compute the average value of xxf )(over the interval 90 x

Using a = 0 and b = 9, the average value of f(x) over the interval is

9

009

1dxx

Then

Consumers’ Surplus

Using a demand curve, we can derive a formula that shows the amount that consumers benefit from an open system that has no price discrimination.

Consider a typical demand curve p = f(x) where price decreases as quantity increases.

Let A designate the amount available and B = f(A) be the current selling price.

Divide the interval from 0 to A into n subintervals and take xi to be the right-hand

endpoint of the ith interval.

Consider the first subinterval from 0 to x1.

Suppose that only x1 units had been available

The price per unit could have been set at f(x1)

and these x1 units sold at that price.

However, at this price no more units could be sold.

Selling the first x1 units at f(x1) would yield

Now, suppose that after selling the first units, more units become available so that there is now a total of x2 units that have been

produced.

If the price is set at f(x2), the remaining

x2-x1 = Δx units can be sold.

Continuing this process of price discrimination, the amount of money paid by consumers would be a Riemann sum

Taking n large, we note that the Riemann sum approaches

A

dxxf0

)(

Since f(x) is positive, this is the area under the graph of f(x) from x = 0 to x = A.

However, in an open system, everyone pays the same price B, so the total amount paid by consumers is

[price per unit][number of units] = BA

BA is the area under the graph of the line p = B from x = 0 to x = A. The amount of money saved by consumers is the area between the curves p = f(x) and p = B.

The consumers’ surplus for a commodity having demand curve p = f(x) is

A

dxBxf0

])([

where the quantity demanded is A and the price is B = f(A).

Find the consumers’ surplus for each of the following demand curves at the given sales level x.

20;50200

2

xxx

p

Future Value of an Income Stream

The future value, of a continuous income stream, of K dollars per year for N years at interest rate r compounded continuously is

N tNr dtKe0

)(

Suppose that money is deposited steadily into a savings account at the rate of $14,000 per year.

Determine the balance at the end of 6 years if the account pays 4.5% interest compounded continuously.