Area

1
Area

• Section 4.2

2
Summation
Sigma is used to denote
summation.   The sum of n terms a1, a2, a3, , an
is expressed as i is called the
______________________________________1 is the
____________________________ n is the
___________________________ .ai is the ith term
of the sum.
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Summation Examples
Example
Example
Example
4
Summation
Example
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Summation Rules
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Summation Rules
7
Examples
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Area
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• Consider the region bounded by the graphs of
• The area can be approximated by two sets of
  rectanglesone set inscribed within the region
  and the other set circumscribed over the region.

The actual area lies between the lower and upper
sums.
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Lower Approximation

• Find the sum of the areas of the inscribed
  rectangles.

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Upper Approximation

• Find the sum of the areas of the circumscribed
  rectangles.

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Continued
The actual area lies between the lower and upper
sums.
Thus, the area bounded by the graphs of is
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Example

• Find the lower and upper approximations of the
  area of the region lying between the graph of
  and the x-axis between x 0
  and x 2. Use 4 rectangles.

1) Lower Sum
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Example

• 2) Upper Sum

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The limit
Notice The smaller the intervals (the
greater the number of rectangles), the
closer the approximate area is to the actual
area.   In fact, the exact area can be found by
Area under curve
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As n increases without bound
As you increase the number of rectangles, the
approximation tends to become better because the
amount of missed area decreases.
Check this out http//xanadu.math.utah.edu/java/A
pproxArea.html
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Homework

• Section 4.2 page 267 1 7 odd, 15, 23, 27, 29