applications of the definite integral math 150 spring 2004
TRANSCRIPT
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Applications of The Definite Integral
Math 150Spring 2004
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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
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Example
Ex: Find the average value of f(x)=3x2 for x=1 to x=3Solution:
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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
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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
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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
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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
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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?
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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
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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: