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# lec23 - .01 Fall 2006 Lecture 23 Work Average Value...

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Unformatted text preview: 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 < x 1 < x n = b · · · y = f ( x ) , y 1 = f ( x 1 ) , ...y n = f ( x n ) and n (Δ 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 dθ not dx . See Figure 3) equal weighting in θ different weighting in x Figure 3: Different weighted averages. 2 Lecture 23 18.01 Fall 2006 y = sin θ π 1 1 (- cos θ ) π 1 2 Average = sin θ dθ = (- cos π- (- cos 0)) = = π π π π Example 4. Find the average temperature of water in the witches cauldron from last lecture. (See Figure 4)....
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lec23 - .01 Fall 2006 Lecture 23 Work Average Value...

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