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lec19 - 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 19 18.01 Fall 2006 Lecture 19: First Fundamental Theorem of Calculus Fundamental Theorem of Calculus (FTC 1) If f ( x ) is continuous and F ( x ) = f ( x ), then b f ( x ) dx = F ( b ) F ( a ) a Notation: F ( x ) b = F ( x ) x = b = F ( b ) F ( a ) a x = a b b 3 3 b 3 a 3 x x 2 ; x 2 dx = Example 1. F ( x ) = F ( x ) = x = 3 3 3 , 3 a a Example 2. Area under one hump of sin x (See Figure 1.) π 0 sin x dx = cos x π = cos π ( cos 0) = ( 1) ( 1) = 2 0 1 Figure 1: Graph of f ( x ) = sin x for 0 x π . 1 1 6 = 1 1 6 0 = 6 5 dx = x Example 3. x 6 0 0 1
Lecture 19 18.01 Fall 2006 Intuitive Interpretation of FTC: dx x ( t ) is a position; v ( t ) = x ( t ) = is the speed or rate of change of x . dt b v ( t ) dt = x ( b ) x ( a ) (FTC 1) a R.H.S. is how far x ( t ) went from time t = a to time t = b (difference between two odometer readings).

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