Integral Calculus

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Integral Calculus

• Chapter 13

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Anti-derivatives

• ? xn dx ((x(n 1))/(n 1)) C
• n ? -1
• ? ekx dx (1/k)ekx C
• k ? 0

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Example
Area under the curve one half of square
(1/2)xx Evaluate for x 2 yields area 2
? xn dx ((x(n 1))/(n 1)) C Evaluate from
x 0 to x 2 and for n 1 ((2(1 1))/(1
1)) C (((0(1 1))/(1 1)) C) (22)/2 2
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Example 1

• When you buy a used car, you may be offered the
  opportunity to buy a warranty that covers repair
  costs for a certain period of time. A company
  that provides such warranties uses current data
  and past experience to determine that the annual
  repair costs for a particular model in a given
  year x by
• r(x) 120e(.4x)

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Example 1

• What will the total repair costs be for one year
  and for three years?

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r(x) 120e(.4x)
What is the area under the curve?
Estimations Year 1 150 Year 3 3400/2
600
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• ? ekx dx (1/k)ekx C
• ? 120e.4x dx 120(1/.4)e.4x C
• Evaluate for x 0 and x 1, then subtract the
  first from the second
• (in this process the constant C will be
  eliminated so we dont need to know it!)

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• ? 120e.4x dx 120(1/.4)e.4x C
• ? 120e.41dx 120(1/.4)e.41 C 300e.4 C
• ? 120e.40 dx 120(1/.4)e.40 C 300e0
  C
• Difference 300e.4 300 147.55

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• ? ekx dx (1/k) ekx C
• ? 120e.4x dx 120(1/.4)e.4x C
• Evaluate for x 0 and x 3, then subtract the
  first from the second

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• ? 120e.4x dx 120(1/.4)e.4x C
• ? 120e.43 dx 120(1/.4)e.43 C 300e1.2
  C
• ? 120e.40 dx 120(1/.4)e.40 C 300e0
  C
• Difference 300e1.2 300 696.04

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Example 2

• According to data from the toy Manufacturers of
  America, the marginal revenue from video game
  sales (in billions of dollars per year) is
  approximately given by
• MR(x) 1.68x 17
• where x 0 corresponds to 1996.

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Example 2

• What was the total video game revenue from 1996
  to 2000?
• Marginal revenue is the derivative of the revenue
  function.

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MR(x) 1.68x 17
Estimation of the area under the curve 417
(rectangle) 424/4 68 24 92
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• Revenue Integral of Marginal Revenue
• ? (1.68x 17) dx
• (1/2)1.68x2 17x
• .84x2 17x
• Evaluate this expression for x 4 and x 0.
  Then find the difference between these
  expressions.
• .8442 174 81.44 billion

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Example 2

• At what point in time did total revenue reach 50
  billion?

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• ? (1.68x 17) dx
• (1/2)1.68x2 17x
• .84x2 17x
• 50 .84x2 17x
• .84x2 17x 50 0
• X 2.6, so after mid-year in 1998