Applications of The Definite Integral

Math 150Spring 2004

Average Value of f (x)

Roughly, the number of values of f(x) from x=a to x=b is (b-a)Average value is then

Example

Ex: Find the average value of f(x)=3x2 for x=1 to x=3Solution:

Future Value of Income

Suppose you deposit $1000 in a savings account every year for 10 years and it earns 5% interest per year compounded continuously. Find the value of each $1000 ( 10 years, 9 years, …) and add them together.This is approximately the future value of your deposits

Each Piece Grows

Summing up each deposit and its interestYears Compounded Starting Value Compounded Value

10 $1,000.00 $1,648.729 $1,000.00 $1,568.318 $1,000.00 $1,491.827 $1,000.00 $1,419.076 $1,000.00 $1,349.865 $1,000.00 $1,284.034 $1,000.00 $1,221.403 $1,000.00 $1,161.832 $1,000.00 $1,105.171 $1,000.00 $1,051.27

Total $13,301.49

More Frequent Deposits

20 deposits of $500

Years Compounded Starting Value Compounded Value10 $500.00 $824.369.5 $500.00 $804.019 $500.00 $784.168.5 $500.00 $764.808 $500.00 $745.917.5 $500.00 $727.507 $500.00 $709.536.5 $500.00 $692.026 $500.00 $674.935.5 $500.00 $658.275 $500.00 $642.014.5 $500.00 $626.164 $500.00 $610.703.5 $500.00 $595.623 $500.00 $580.922.5 $500.00 $566.572 $500.00 $552.591.5 $500.00 $538.941 $500.00 $525.640.5 $500.00 $512.66

Total $13,137.28

The Formula

As the number of deposits per year increases, this sum becomes an integralThe future value of $P deposited per year for N years with interest continuously compounded at a rate r per year is

ExampleSuppose you deposit $1000 per year in an account bearing 5% interest per year compounded continuously. What is the future value of this account in 10 years?

Present Value of Income

On the other hand, you can ask “How much are payments worth today?” if they are paid over N years.A deposit or payment of P dollars made t years from now will have value Pe-rt

Adding these up over N years gives

Example

What is the present value of payments made continuously for 10 years at $1000 per year if interest rates are 5% compounded continuously?Solution: