05-06 - DefInt

05-06 - DefInt - Definite Integrals We have approximated...

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Definite Integrals We have approximated the area under a curve by using rectangles. The more rectangles, the better the approximation. The exact area is lim n !" f x i ( ) i = 1 n # $ x where ! x is the width (base) of each rectangle and f x i ( ) is the height of the ith rectangle. Definition: If f is a continuous function on [a, b], we divide [a, b] into n subintervals of equal width. ! x = b " a n The definite integral of f from a to b is f ( x ) dx a b ! = lim n "# f x i ( ) i = 1 n $ % x where x i is some point in the ith subinterval.
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i ( ) i = 1 n ! " x is a Riemann sum. If we choose the left hand endpoint, we have a left Riemann sum. If we choose the right hand endpoint, we have a right Riemann sum. Note: 1. The definite integral is a number. 2. It represents area under a curve f(x) only if f(x) 0 on [a, b]. If f(x) < 0 somewhere on the interval, the integral does not represent area. Ex.
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This note was uploaded on 10/16/2009 for the course MATH 1206 taught by Professor Llhanks during the Spring '08 term at Virginia Tech.

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05-06 - DefInt - Definite Integrals We have approximated...

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