Fundamental Theorem of Calculus

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Fundamental Theorem of Calculus

• Section 4.4

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Fundamental Theorem of Calculus

• The Fundamental Theorem of CalculusIf a
  function f is continuous on the closed interval
  a, b and F is an antiderivative of f on the
  interval a, b then
•  

We express this as
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Examples
Example
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Examples
Example
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Examples
Example
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Examples
Example
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Examples
Example
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Examples
Find the area bounded by the graphs of y x
sin x, the x-axis, x 0, and x
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Mean Value Theorem for Integrals
If f is continuous on a, b, then there exists a
number c in the closed interval a, b such that

Somewhere between the inscribed and circumscribed
rectangles there is a rectangle whose area is
precisely equal to the area under the curve.
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Average Value of a Function

• The value of f(c) in the Mean Value Theorem for
  Integrals is called the average value of f on the
  interval a, b.
•  
• Since then solving
  for f(c) gives

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Example

• Find the average value of f(x) sin x on 0, ?
  and all values of x for which the function equals
  its average value.

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The Second Fundamental Theorem of Calculus
Using x as the upper limit of integration.
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Example

• Integrate to find F as a function of x and
  demonstrate the Second Fundamental Theorem of
  Calculus by differentiating the result.
•  

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Example

• Integrate to find F as a function of x and
  demonstrate the Second Fundamental Theorem of
  Calculus by differentiating the result.
•  

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Homework

• Sect 4.4 page 291 5 23 odd, 27, 29, 33 39
  odd, 47