6.5 Applications of the Definite Integral

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6.5 Applications of the Definite Integral
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• In this section, we will introduce applications
  of the definite integral.
• Average Value of a Function
• Consumers Surplus
• Future Value of an Income Stream

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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
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Compute the average value of
over the interval
Using a 0 and b 9, the average value of f(x)
over the interval is
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Then
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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.
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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.
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Divide the interval from 0 to A into n
subintervals and take xi to be the right-hand
endpoint of the ith interval.
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Consider the first subinterval from 0 to x1.
Suppose that only x1 units had been
available The price per unit could have been set
at f(x1) and these x1 units sold at that price.
However, at this price no more units could be
sold. Selling the first x1 units at f(x1) would
yield
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Now, suppose that after selling the first units,
more units become available so that there is now
a total of x2 units that have been produced.
If the price is set at f(x2), the remaining
x2-x1 ?x units can be sold. Continuing this
process of price discrimination, the amount of
money paid by consumers would be a Riemann sum
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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.
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However, in an open system, everyone pays the
same price B, so the total amount paid by
consumers is price per unitnumber of units
BA
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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.
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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).
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Find the consumers surplus for each of the
following demand curves at the given sales level
x.
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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
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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.