Business and Economic Applications
Objectives:
Students will be able to
 Apply integrals to business and economic application problems.
The definite integral and its concept of area under the curve have many applications
in statistics, business, and economics. The application that we will focus on here has
to do with money and is referred to as the continuous money flow.
Here we will use concepts of continuously compounded interest and present value
to find total money flow, present value of money flow, and accumulated amount of
money flow at time t.
Total money flow is basically total income. If the amount of income is constant over
time, then calculating total income is basically a geometric problem of finding the
area of a rectangle. If income changes at a given rate, then we have the following
If f(x) is the rate of money flow, the then total money flow over the time interval
x = 0 to x = t is given by

t
0
f ( x)dx .
The present value of an investment is the amount of money that would need to be
invested now at a prescribed interest rate for the set amount of time to reach a
future. To find the present value of money flow, we will use the
certain value in the
rate of income and the continuous compound interest formula.
If f(x) is the rate of continuous money flow at an interest rate r for t years, then
the present value is P 

t
0
f ( x)erx dx .
To find the accumulated amount of money flow with interest at time t, we will use
the continuous compound interest formula A  Pert and our formula for present

value of money flow.
If f(x) is the rate of money flow at an interest rate r at time x, the accumulated

t
amount of money flow at time t is A  e rt  f ( x)erx dx .
0
Example 1:
The function f ( x) 1100e0.07x represents the rate of flow of money in dollars per

year. Assume a 15-year period at 8% compounded continuously. Find the present
value and the accumulated amount of money flow at t = 15.
