# lec18 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Lecture 18 18.01 Fall 2006 Lecture 18: Defnite Integrals Integrals are used to calculate cumulative totals, averages, areas. Area under a curve : (See Figure 1.) 1. Divide region into rectangles 2. Add up area of rectangles 3. Take limit as rectangles become thin a b a b (i) (ii) Figure 1: (i) Area under a curve; (ii) sum of areas under rectangles Example 1. f ( x ) = x 2 , a = 0, b arbitrary 1. Divide into n intervals Length b/n = base of rectangle 2. Heights: ± 2 b b 1 st : x = , height = n n ± 2 2 b 2 b 2 nd : x = , height = n n Sum of areas of rectangles: ± ± 2 ± ± 2 ± ± 2 ± ± 2 b b b 2 b b 3 b b nb b 3 + + + + = (1 2 + 2 2 + 3 2 + + n 2 ) n n n n n n ··· n n n 3 ··· 1
Lecture 18 18.01 Fall 2006 a=0 b 2 Figure 2: Area under f ( x ) = x above [0 ,b ]. We

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## This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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lec18 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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