Applications of The Definite Integral Math 150 Spring 2004.

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Presentation transcript:

Applications of The Definite Integral Math 150 Spring 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)=3x 2 for x=1 to x=3 Solution:

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 interest

More Frequent Deposits 20 deposits of $500

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

Example Suppose 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: