Explanation Given: The functions f ( x ) = x sin 2 x and g ( x ) = x 2 sin 2 x . The intervals [ 0 , 1 ] and [ 1 , 2 ] . Formula used: Average value of a function f ( x ) over an interval [a, b] is given by 1 b − a ∫ a b f ( x ) d x ∫ u v d x = u ∫ v d x − ∫ u ′ ( ∫ v d x ) d x Calculation: Let us find the average value of the given functions over the intervals [ 0 , 1 ] and [ 1 , 2 ] . Now, Calculating the average value of given function, f ( x ) = x sin 2 x over [ 0 , 1 ] , We obtain, Average value = 1 1 − 0 ∫ 0 1 x sin 2 x d x = ∫ 0 1 x sin 2 x d x Evaluating ∫ 0 1 x sin 2 x d x by using Integration by parts formula for Integration;. u = m ( x ) , v = h ( x ) ∫ x sin 2 x d x = x ∫ sin 2 x d x − ∫ d d x ( x ) ( ∫ sin 2 x ) d x = x ∫ 1 − cos 2 x 2 d x − ∫ 1. ( ∫ 1 − cos 2 x 2 ) d x = x 2 ( x 2 − sin 2 x 2 ) − ∫ ( x − sin 2 x 2 ) d x = x 2 ( x − sin 2 x 2 ) − 1 2 ( x 2 2 + cos 2 x 4 ) + C Computing the boundaries for the above evaluated integral over [ 0 , 1 ] , we obtain, ∫ 0 1 x sin 2 x d x = ( 1 4 ) − ( sin 2 4 ) − ( cos 2 8 ) + ( 1 8 ) =0.19969. Computing the boundaries for ∫ x sin 2 x d x over [ 1 , 2 ] , we obtain, ∫ 1 2 x sin 2 x d x = [ 2 2 ( 2 − sin 4 2 ) − 1 2 ( 4 2 + cos 4 2 ) ] - [ 1 2 ( 1 − sin 2 2 ) − 1 2 ( 1 2 + cos 2 4 ) ] Simplifying the above expression for ∫ 1 2 x sin 2 x d x , we obtain the average value over [ 1 , 2 ] as 1.38541. Hence, the average value of ∫ x sin 2 x d x over [ 0 , 1 ] is 0...

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Chapter 6.2, Problem 63E

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Which of the given two functions has larger average value over the given two intervals?

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Chapter 6.2, Problem 62E

Chapter 6.2, Problem 64E

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Chapter 6 Solutions

CALCULUS (CLOTH)

Ch. 6.1 - Prob. 1PQ Ch. 6.1 - Prob. 2PQ Ch. 6.1 - Prob. 3PQ Ch. 6.1 - Prob. 4PQ Ch. 6.1 - Prob. 5PQ Ch. 6.1 - Prob. 6PQ Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E

Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - Prob. 13E Ch. 6.1 - Prob. 14E Ch. 6.1 - Prob. 15E Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Prob. 39E Ch. 6.1 - Prob. 40E Ch. 6.1 - Prob. 41E Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - Prob. 47E Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Prob. 53E Ch. 6.1 - Prob. 54E Ch. 6.1 - 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Prob. 66E Ch. 6.4 - Prob. 1PQ Ch. 6.4 - Prob. 2PQ Ch. 6.4 - Prob. 3PQ Ch. 6.4 - Prob. 1E Ch. 6.4 - Prob. 2E Ch. 6.4 - Prob. 3E Ch. 6.4 - Prob. 4E Ch. 6.4 - Prob. 5E Ch. 6.4 - Prob. 6E Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 8E Ch. 6.4 - Prob. 9E Ch. 6.4 - Prob. 10E Ch. 6.4 - Prob. 11E Ch. 6.4 - Prob. 12E Ch. 6.4 - Prob. 13E Ch. 6.4 - Prob. 14E Ch. 6.4 - Prob. 15E Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 19E Ch. 6.4 - Prob. 20E Ch. 6.4 - Prob. 21E Ch. 6.4 - Prob. 22E Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - Prob. 25E Ch. 6.4 - Prob. 26E Ch. 6.4 - Prob. 27E Ch. 6.4 - Prob. 28E Ch. 6.4 - Prob. 29E Ch. 6.4 - Prob. 30E Ch. 6.4 - Prob. 31E Ch. 6.4 - Prob. 32E Ch. 6.4 - Prob. 33E Ch. 6.4 - Prob. 34E Ch. 6.4 - Prob. 35E Ch. 6.4 - Prob. 36E Ch. 6.4 - Prob. 37E Ch. 6.4 - Prob. 38E Ch. 6.4 - Prob. 39E Ch. 6.4 - Prob. 40E Ch. 6.4 - Prob. 41E Ch. 6.4 - Prob. 42E Ch. 6.4 - Prob. 43E Ch. 6.4 - Prob. 44E Ch. 6.4 - Prob. 45E Ch. 6.4 - Prob. 46E Ch. 6.4 - Prob. 47E Ch. 6.4 - Prob. 48E Ch. 6.4 - Prob. 49E Ch. 6.4 - Prob. 50E Ch. 6.4 - Prob. 51E Ch. 6.4 - Prob. 52E Ch. 6.4 - Prob. 53E Ch. 6.4 - Prob. 54E Ch. 6.4 - Prob. 55E Ch. 6.4 - Prob. 56E Ch. 6.4 - Prob. 57E Ch. 6.4 - Prob. 58E Ch. 6.4 - Prob. 59E Ch. 6.4 - Prob. 60E Ch. 6.4 - Prob. 61E Ch. 6.4 - Prob. 62E Ch. 6.4 - Prob. 63E Ch. 6.4 - Prob. 64E Ch. 6.5 - Prob. 1PQ Ch. 6.5 - Prob. 2PQ Ch. 6.5 - Prob. 3PQ Ch. 6.5 - Prob. 4PQ Ch. 6.5 - Prob. 1E Ch. 6.5 - Prob. 2E Ch. 6.5 - Prob. 3E Ch. 6.5 - Prob. 4E Ch. 6.5 - Prob. 5E Ch. 6.5 - Prob. 6E Ch. 6.5 - Prob. 7E Ch. 6.5 - Prob. 8E Ch. 6.5 - Prob. 9E Ch. 6.5 - Prob. 10E Ch. 6.5 - Prob. 11E Ch. 6.5 - Prob. 12E Ch. 6.5 - Prob. 13E Ch. 6.5 - Prob. 14E Ch. 6.5 - Prob. 15E Ch. 6.5 - Prob. 16E Ch. 6.5 - Prob. 17E Ch. 6.5 - Prob. 18E Ch. 6.5 - Prob. 19E Ch. 6.5 - Prob. 20E Ch. 6.5 - Prob. 21E Ch. 6.5 - Prob. 22E Ch. 6.5 - Prob. 23E Ch. 6.5 - Prob. 24E Ch. 6.5 - Prob. 25E Ch. 6.5 - Prob. 26E Ch. 6.5 - Prob. 27E Ch. 6.5 - Prob. 28E Ch. 6.5 - Prob. 29E Ch. 6.5 - Prob. 30E Ch. 6.5 - Prob. 31E Ch. 6.5 - Prob. 32E Ch. 6.5 - Prob. 33E Ch. 6.5 - Prob. 34E Ch. 6.5 - Prob. 35E Ch. 6.5 - Prob. 36E Ch. 6.5 - Prob. 37E Ch. 6.5 - Prob. 38E Ch. 6.5 - Prob. 39E Ch. 6.5 - Prob. 40E Ch. 6.5 - Prob. 41E Ch. 6.5 - Prob. 42E Ch. 6.5 - Prob. 43E Ch. 6 - Prob. 1CRE Ch. 6 - Prob. 2CRE Ch. 6 - Prob. 3CRE Ch. 6 - Prob. 4CRE Ch. 6 - Prob. 5CRE Ch. 6 - Prob. 6CRE Ch. 6 - Prob. 7CRE Ch. 6 - Prob. 8CRE Ch. 6 - Prob. 9CRE Ch. 6 - Prob. 10CRE Ch. 6 - Prob. 11CRE Ch. 6 - Prob. 12CRE Ch. 6 - Prob. 13CRE Ch. 6 - Prob. 14CRE Ch. 6 - Prob. 15CRE Ch. 6 - Prob. 16CRE Ch. 6 - Prob. 17CRE Ch. 6 - Prob. 18CRE Ch. 6 - Prob. 19CRE Ch. 6 - Prob. 20CRE Ch. 6 - Prob. 21CRE Ch. 6 - Prob. 22CRE Ch. 6 - Prob. 23CRE Ch. 6 - Prob. 24CRE Ch. 6 - Prob. 25CRE Ch. 6 - Prob. 26CRE Ch. 6 - Prob. 27CRE Ch. 6 - Prob. 28CRE Ch. 6 - Prob. 29CRE Ch. 6 - Prob. 30CRE Ch. 6 - Prob. 31CRE Ch. 6 - Prob. 32CRE Ch. 6 - Prob. 33CRE Ch. 6 - Prob. 34CRE Ch. 6 - Prob. 35CRE Ch. 6 - Prob. 36CRE Ch. 6 - Prob. 37CRE Ch. 6 - Prob. 38CRE Ch. 6 - Prob. 39CRE Ch. 6 - Prob. 40CRE Ch. 6 - Prob. 41CRE Ch. 6 - Prob. 42CRE Ch. 6 - Prob. 43CRE Ch. 6 - Prob. 44CRE Ch. 6 - Prob. 45CRE Ch. 6 - Prob. 46CRE Ch. 6 - Prob. 47CRE Ch. 6 - Prob. 48CRE Ch. 6 - Prob. 49CRE Ch. 6 - Prob. 50CRE Ch. 6 - Prob. 51CRE Ch. 6 - Prob. 52CRE Ch. 6 - Prob. 53CRE Ch. 6 - Prob. 54CRE

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Which of the given two functions has larger average value over the given two intervals?

Solution Summary: The author calculates the average value of the given functions over the intervals [0, 1] and [1, 2].

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Which of the given two functions has larger average value over the given two intervals?

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Chapter 6 Solutions