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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.

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