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Unformatted text preview: MATH 006 Calculus and Linear Algebra (Lecture 32) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 18 Outline 1 Area under a graph 2 Definite Integrals 3 Definite integrals and Substitution 4 Average Value of a Continuous Function Albert Ku (HKUST) MATH 006 2 / 18 Area under a graph Area under a graph Question Given the graph of y = f ( x ), as shown in the figure below. How can we find the area S under the graph for a ≤ x ≤ b ? Albert Ku (HKUST) MATH 006 3 / 18 Area under a graph Riemann Sum Idea: Approximate S by rectangles! The sum of area of all such rectangles is called the Riemann sum . However, Riemann sum is just an approximation. How can we find the exact value of S ? Albert Ku (HKUST) MATH 006 4 / 18 Area under a graph “Limit” of Riemann Sum Another idea: Narrower rectangles give better approximation! Suppose y = f ( x ) = x 2 . We want to find the area under this graph for ≤ x ≤ 1. View this Java applet for details. The “limit” of Riemann sum is the area S ! Albert Ku (HKUST) MATH 006 5 / 18 Area under a graph Signed Area What is the limit of Riemann sum if the graph is not lying entire above the xaxis? The green area below the xaxis will be counted as “negative area” in the limit of Riemann sum. Hence, we have The limit of Riemann sum = signed area = Yellow area  Green area Albert Ku (HKUST) MATH 006 6 / 18 Definite Integrals Definite Integrals Definition The definite integral of f ( x ) from a to...
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This note was uploaded on 12/28/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.
 Fall '09
 forgot
 Math, Linear Algebra, Algebra, Integrals

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