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# lec23 - 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 23 18.01 Fall 2006 Lecture 23: Work, Average Value, Probability Application of Integration to Average Value You already know how to take the average of a set of discrete numbers: a 1 + a 2 a 1 + a 2 + a 3 or 2 3 Now, we want to find the average of a continuum. y=f(x) a b . x 4 y 4 . Figure 1: Discrete approximation to y = f ( x ) on a x b . Average y 1 + y 2 + ... + y n n where a = x 0 < x 1 < x n = b · · · y 0 = f ( x 0 ) , y 1 = f ( x 1 ) , . . . y n = f ( x n ) and n x ) = b a ⇐⇒ Δ x = b n a and The limit of the Riemann Sums is b lim ( y 1 + · · · + y n ) b n a = f ( x ) dx a n →∞ Divide by b a to get the continuous average y 1 + + y n 1 b lim · · · = f ( x ) dx n →∞ n b a a 1
Lecture 23 18.01 Fall 2006 area = /2 y=√ 1-x 2 Figure 2: Average height of the semicircle. Example 1. Find the average of y = 1 x 2 on the interval 1 x 1. (See Figure 2) 1 1 1 π π Average height = 2 1 x 2 dx = 2 2 = 4 1 Example 2. The average of a constant is the same constant b 1 53 dx = 53 b a a Example 3. Find the average height y on a semicircle, with respect to arclength . (Use not dx . See Figure 3) equal weighting in θ different weighting in x Figure 3: Different weighted averages. 2

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Lecture 23 18.01 Fall 2006 y = sin θ π 1 1 ( cos θ ) π 1 2 Average = sin θ = ( cos π ( cos 0)) = = 0 π π π π 0 Example 4. Find the average temperature of water in the witches cauldron from last lecture. (See Figure 4). 2m 1m Figure 4: y = x 2 , rotated about the y -axis. First, recall how to find the volume of the solid of revolution by disks.
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lec23 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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