5.3: The Fundamental Theorem of Calculus

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5.3 The Fundamental Theorem of Calculus
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The Fundamental Theorem of Calculus

• Let f be a continuous function, defined on an
  open interval I containing a. The function Af
  with the rule
• is defined for every x in I and

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• Let f(x) 3x2 cos(x3) and .
• Determine a symbolic representation for Af
  (including C).

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Graph f(x) and Af(x) on the same grid. Use your
graph to Determine if your result is reasonable.
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Remember

• Suppose that F(x) G(x) C for all x in an
  interval I. Then for some constant C, F(x)
  G(x) C for all x in I.

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Determine
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Determine
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The Fundamental Theorem with the Chain Rule

• If either limit of integration is a function of x
  rather than just x, we must use the chain rule
  when applying the FTC. In other words (or
  symbols)

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(No Transcript)
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Determine
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Determine
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Determine
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Alternative Version of the FTC

• Let f be continuous on a,b and let F be any
  anti-derivative of f. Then,

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A bit o shorthand

• Instead of writing out
• all of the time, we usually write

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• Use the alternative form of the FTC to
• Interpret the integral as an area.

determine
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Evaluate the integral using the Alternative Form
of the FTC
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Evaluate the integral using the Alternative Form
of the FTC
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• Determine the area under the graph of f(x)
  1/x from 1 to b. Which value of b makes the area
  1?

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Indefinite and Definite Integrals

• So far, we have been using definite integrals
  (i.e. integrals with given limits of
  integration). An indefinite integral looks just
  like a definite integral, but without limits of
  integration.

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Definite and Indefinite Integrals
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• Definite integrals represent a specific number
  whereas an indefinite integral represents an
  entire family of functions (allowing for multiple
  values of C).

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Evaluate to Illustrate the point of the last slide
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• Remember the FTC only works when we have an
  anti-derivative for our given function.

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• Which of these functions gives the signed area
  under the curve y t2 between t 2 and t x as
  a function of x?
• A) B) C)
• D) E) none of these

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1. 1/2 sin x2 - ½ sin (-172)
2. x cos (x2) (-17)cos (-172)
3. 1/2 sin x2
4. x cos (x2)
5. None of these

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1. 160/3
2. 52
3. 128/3
4. 56
5. None of these

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Class Work

• pages 374 and 375,
• 1 and 3

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Assignment

• pages 374 376,
• 4, 7, 10 and 11