lec18 - Lecture 18 18.01 Fall 2006 Lecture 18: Definite...

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Unformatted text preview: Lecture 18 18.01 Fall 2006 Lecture 18: Definite 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 will now estimate the sum using some 3-dimensional geometry. Consider the staircase pyramid as pictured in Figure 3. n = 4 n Figure 3: Staircase pyramid: left(top view) and right (side view) 1 st level: n n bottom, represents volume...
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lec18 - Lecture 18 18.01 Fall 2006 Lecture 18: Definite...

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