The Area Between Two Curves 7'5

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The Area Between Two Curves7.5

• JMerrill, 2007

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What You Already Know

• Many important integrals require finding the area
  between 2 graphs . We have looked at finding the
  area between a graph and the x-axis. Now it will
  be between 2 graphs--

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Area Between Two Curves

• If f and g are continuous functions and f(x)
  g(x) from xa to xb on a,b nthen the area
  between the curves is given by

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Area

• Find the area bounded by f(x) -x21, g(x)2x4,
  x-1, and x2

g(x)
f(x)
Notice the graphs do not intersect and g is on top
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Area

• Find the area between y x ½ and y x3

g(x)
f(x)
These graphs intersect so you must find the
intersection pointsset them to each other and
solve or do it on the calculator
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Calculator Applications

• fnInt on Math Menu
• fnInt(expression, variable, lower limit, upper
  limit)
• Last problem fnInt(x.5-x3,x,0,1) ENTER
• .416666667 5/12
• Use this to check your answers. Next area
  problem on board.

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AreaYou Do

• A company is considering a new manufacturing
  process in one of its plants. The new process
  provides substantial initial savings, with the
  savings declining with time x (in years)
  according to the rate-of-savings function
  S(x)100 - x2 where s(x) is in thousands of per
  year
• At the same time, the cost of operating the new
  process increases with time x (in years),
  according to the rate-of-cost function (in
  thousands of per year) C(x) x2 14/3 x

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Area-You Do
S(x)100 - x2
The company will save 372,000 over the 6-year
period
(6,64)
C(x) x2 14/3 x
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Consumers Surplus

• The market determines the price at which a
  product is sold. The demand curve and the supply
  curve meet at the equilibrium price. At that
  price, consumers will purchase the same amount of
  a product, q0, that manufacturers want to sell.
• However, some consumers would be willing to spend
  more. The total differences between the
  equilibrium price, p0, and the higher price is
  the consumers surplus.

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Producers Surplus

• If some manufacturers supply a product at a lower
  than equilibrium price, p0, the total differences
  between the equilibrium price and the lower price
  is the producers surplus.

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Consumer Producer Surplus

• Consumers Surplus
• Producers Surplus

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Problem

• Suppose the price ( per ton) for oat bran is
  when the demand for the
  product is q tons
• Suppose the function gives the
  price ( per ton) when the supply is q tons.
• Find the consumers surplus and the producers
  surplus.

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Problem
(10.60, 199.5)