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 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 is defined as the quantity

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

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 x i to be the right-hand endpoint of the ith interval.

Consider the first subinterval from 0 to x 1. Suppose that only x 1 units had been available The price per unit could have been set at f(x 1 ) and these x 1 units sold at that price. However, at this price no more units could be sold. Selling the first x 1 units at f(x 1 ) would yield

Now, suppose that after selling the first units, more units become available so that there is now a total of x 2 units that have been produced. If the price is set at f(x 2 ), the remaining x 2 -x 1 = 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 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 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.

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

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.