AREA

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AREA

• Section 4.2

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When you are done with your homework, you should
be able to

• Use sigma notation to write and evaluate a sum
• Understand the concept of area
• Approximate the area of a plane region
• Find the are of a plane region using limits

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SIGMA NOTATION

• The sum of n terms is written as
  
• where i is
  the index of summation, is the ith term of
  the sum, and the upper and lower bounds of
  summation are n and 1.
• The lower bound can be any number less than or
  equal to the upper bound

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Evaluate

• 25/12
• 0.0

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Evaluate

• 70
• 0.0

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Summation Properties

• 1.
• 2.

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Theorem Summation Formulas

• 1.
• 2.
• 3.
• 4.

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AREA

• Recall that the definition of the area of a
  rectangle is
• From this definition, we can develop formulas for
  many other plane regions

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AREA continued

• We can approximate the area of f by summing up
  the areas of the rectangles
• What happens to the area approximation when the
  width of the rectangles decreases?

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Theorem Limits of the Lower and Upper Sums

• Let f be continuous and nonnegative on the
  interval . The limits as of
  both the lower and upper sums exist and are equal
  to each other. That is,
• Where and and
  are the minimum and maximum values of f on the
  subinterval.

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Continued

• Since the same limit is attained for both the
  minimum value and the maximum, it follows from
  Squeeze Theorem that the choice of x in the ith
  subinterval does not affect the limit. This
  means that we can choose an arbitrary x-value in
  the ith subinterval.

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Definition of an Area in the Plane

• Let f be continuous and nonnegative on the
  interval . The area of the region bounded
  by the graph of f, the x-axis, and the vertical
  lines and is

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Find the area of the region bounded by the graph
, the x-axis, and the vertical lines and
.

• 0.25
• 0.0