Sigma Notation, Upper and Lower Sums

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Area

• Sigma Notation, Upper and Lower Sums

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Sigma Notation

• Definition a concise notation for sums.
• This notation is called sigma notation because it
  uses the uppercase Greek letter sigma, written as
  ?.
• The sum of n terms

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Examples of Sigma Notation
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Examples of Sigma Notation
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Examples of Sigma Notation
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Summation Formulas
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Using Formulas to Evaluate a Sum

• Evaluate the following summation for n 10, 100,
  1000 and 10,000.

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Using Formulas to Evaluate a Sum

• Now we have to substitute 10, 100, 1000, and
  10,000 in for n.
• n 10 the answer is 0.65000
• n 100 the answer is 0.51500
• n 1000 the answer is 0.50150
• n 10,000 the answer is 0. 50015
• What does the answer appear to approach as the
  ns get larger and larger (limit as n approaches
  infinity)?

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Area

• Finding the area of a polygon is simple because
  any plane figure with edges can be broken into
  rectangles and triangles.
• Finding the area of a circular object or curve is
  not so easy.
• In order to find the area, we break the figure
  into rectangles. The more rectangles, the more
  accurate the area will be.

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Approximating the Area of a Plane Region

• Use five rectangles to find two approximations of
  the area of the region lying between the graph of
• and the x-axis between the graph of x 0 and x
  2.

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Steps

• 1. Draw the graph
• 2. Find the width of each rectangle by taking the
  larger number and subtracting the smaller number.
  Then divide by the number of rectangles
  designated.
• 3. Now find the height by putting the x values
  found in number 2 into the equation.
• 4. Multiply the length times the height (to find
  the area of each rectangle).
• 5. Add each of these together to find the total
  area.

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Approximating the Area of a Plane Region
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Approximating the Area of a Plane Region
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Approximating the Area of a Plane Region

• Now lets find the area using the left endpoints.
  The five left endpoints will involve using the i
  1 rectangle. This answer will be too large
  because there is lots of area being counted that
  is not included (look at the graph).

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Approximating the Area of a Plane Region
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Approximating the Area of a Plane Region

• The true area must be somewhere between these two
  numbers.
• The area would be more accurate if we used more
  rectangles.
• Lets use the program from yesterday to find the
  area using 10 rectangles, 100 rectangles, and
  1000 rectangles.
• What do you think the true area is?

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Upper and Lower Sums

• An inscribed rectangle lies inside the ith region
• A circumscribed rectangle lies outside the ith
  region
• An area found using an inscribed rectangle is
  smaller than the actual area
• An area found using a circumscribed rectangle is
  larger than the actual area
• The sum of the areas of the inscribed rectangles
  is called a lower sum.
• The sum of the areas of the circumscribed
  rectangles is called an upper sum.

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Example of Finding Upper and Lower Sums

• Find the upper an lower sums for the region
  bounded by the graph of
• Remember to first draw the graph.
• Next find the width using the formula

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Example of Finding Lower and Upper Sums
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Example of Finding Lower Sum
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Example of Finding Lower Sum
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Finding an Upper Sum

• Using right endpoints

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Limit of the Lower and Upper Sums

• Let f be continuous and nonnegative on the
  interval
• a, b. The limits as n 8 of both upper and
  lower sums exist and are equal to each other.
  That is,

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Definition of the Area of a Region in the Plane

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