CAB-Chp-5ishC-NTs - CalcAB Chp5ishC Riemann Sums Method...

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Chp5ishC Riemann Sums Method NOTES Produces an approximation for the definite integral 1. NEW METHOD FOR DETERMINING A DEFINITE INTEGRAL (an approximate value) Given a function over the closed interval [a, b] that does not produce a region where geometry area formulas can be used it is possible to approximate its definite integral by using a method derived mostly by a mathematician named Riemann, and his method is known as Riemann's Sum. (Shocking name, hunh?!) In the area under the curve draw rectangles. Get the area of each rectangle. Get the sums of the areas. How to draw the rectangles under the curve can vary. Look at BOTH pictures, notice HOW the rectangles fit under the curve changes; sometimes they can be too short or too tall. For this example five rectangles are drawn. Left-Endpoint Right-Endpoint First left endpoint starts at x=a Last right endpoint finishes at x=b Here the rectangles are too short Here the rectangles are too tall The LEFT endpoint of each rectangle is on f(x) The RIGHT endpoint of each rectangle is on f(x) The area is a small approximation – LOWER SUM The area is a large approximation – UPPER SUM RIEMANN SUMS is a method used to calculate an approximation of the area under a curve by adding the areas of the rectangles formed. The more rectangles drawn the better the approximation. NOTE: Which endpoint (right or left) is used does not strictly determine whether area approximated will be small or large. It depends on the concavity of the curve of the interval used. For example, for a section of graph that is concave down where f(x) is increasing the left- endpoint rectanglesmake alower sum approximation of the area. But, for a section of graph that is concave up where f(x) is decreasing the left-endpoint rectangles makeanuppersum approximation of the area. f(x)
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This note was uploaded on 02/21/2011 for the course MATH 408C taught by Professor Knopf during the Spring '10 term at University of Texas at Austin.

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CAB-Chp-5ishC-NTs - CalcAB Chp5ishC Riemann Sums Method...

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