Riemann Sums

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Riemann Sums
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Objectives

• Students will be able to
• Calculate the area under a graph using
  approximation with rectangles.
• Calculate the area under a graph using geometric
  formulas.

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Riemann Sums
For f(x) a continuous function on the interval
, the area bounded by the
graph of f(x), the x-axis, a, and b using Riemann
sums can be represented by
where and is the x-value in
the ith subinterval so that touches the
graph. As n approaches infinity, this can be
represented as the definite integral
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Riemann Sums
As n approaches infinity, this can be represented
as the definite integral
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Example 1

• Find for the graph of f(x)
  shown below

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Example 2

• Approximate the area under the graph of

and above the x-axis from x 1 to x 9 using
rectangles with n 4 for each of the following
methods
a. left endpoints b. right endpoints c. average
the answers to parts a and b d. midpoints
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Example 3

• Find the exact value of the integral
• using formulas from geometry.

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Example 4

• Find the exact value of the integral
• using formulas from geometry.