Review: 5'3 The Definite Integral

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Review 5.3 The Definite Integral

• Limits of Riemann Sums
• Definition
• Let f be a function defined on a, b. If the
  limit of the Riemann sum is
  I, then I is called the definite integral of f
  over a, b. And we say that f is integrable over
  a, b.
• 2) Notation

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Theorem

• 3) Existence of the Definite Integral
• A continuous function is integrable. That is, if
  a function f is continuous on a, b, then its
  definite integral over a, b exists.

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Properties of Definite Integrals

• 4) Let f(x) and g(x) be integrable funtions on
  a, b. Then
• a)
• b)
• c)
• d)

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Area Under the Graph of a Nonnegative Function

• 2. Area Under a Curve as a definite Integral
• If f(x) is nonnegative and integrable on a, b,
  then the area under the curve of f(x) over a, b
  is the definite integral of f from a to b.

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Example

• Ex.1

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Example

• Ex. 2

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5.4 The Fundamental Theorem of Calculus

• Mean Value Theorem for Definite Integral
• If f is continuous on a, b, then at some point
  c in a, b,
• f(c) is call the average value for f.

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• 2. Fundamental Theorem of Calculus, Part I
• If f is continuous on a, b then
• Is continuous on a, b and differentiable on (a,
  b) and its derivative is f(x).
• That is

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Examples

• Find the derivative.
• a)
• x21
• b)
• sin X

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Examples

• c)
• (x61)2x

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• 3. Fundamental Theorem of Calculus, Part I
• 1) Theorem If f is continuous on a, b and F is
  an antiderivative of f, then

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• 2) Guidelines to use the theorem
• To calculate the definite integral of f over
• a, b,
• Find an antiderivative F of f
• Calculate F(b) and F(a) then do the subtraction.
• It is not necessary to include C in the
  antiderivative.

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Examples

• 3) Examples
• Evaluate each of the following integrals.
• a)
• b)

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Total Area

• 4. Area of Region bounded the graph of a function
  f(x) and x-axis over a, b.
• If f(x) is nonnegative, then the area is given by
  the definite integral
• Otherwise, break the interval into subintervals
  on which the function does not change sign. The
  total area is obtained by adding the absolute
  value of the definite integral over each
  subinterval.

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Examples

• Find the area of the region bounded the graph of
  f(x)x21 over 0,2
• Solution

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• b) Find the area of the region bounded the graph
  of f(x)x2-3x2 over 0,2.
• Solution set x2-3x20, (x-1)(x-2)0,
• x1, x2.

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• So the area of the region under the x-axis is
• The total area is

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Practice

• Page 365
• 2, 4, 6, 8, 12, 44.

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Homework

• 3-11 odd, 19, 23, 27, 31, 37, 41, 43, 63 ON PAGE
  365-367.