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lecture32hout

# lecture32hout - MATH 006 Calculus and Linear...

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MATH 006 Calculus and Linear Algebra (Lecture 32) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 18

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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

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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 0 x 1. View this Java applet for details. The “limit” of Riemann sum is the area S ! Albert Ku (HKUST) MATH 006 5 / 18

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Area under a graph Signed Area What is the limit of Riemann sum if the graph is not lying entire above the x -axis? The green area below the x -axis 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 b

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lecture32hout - MATH 006 Calculus and Linear...

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