Explanation Given Information: The function is f ( x ) = sin x . Formula used: The formula of the average of the function, f ¯ = 1 b − a ∫ a b f ( x ) d x , where a and b is an interval [ a , b ] . Calculation: Consider the provided function, f ( x ) = sin x , To evaluate the average of the function f ( x ) = sin x over the interval [ 0 , π ] , apply the formula f ¯ = 1 b − a ∫ a b f ( x ) d x , Substitute a = 0 , b = π in the formula. f ¯ = 1 π − 0 ∫ 0 π sin x d x Now, evaluate the integral of 1 π ∫ 0 π sin x d x . 1 π ∫ 0 π sin x d x = 1 π [ − cos x ] 0 π Now evaluate the value of integral at the upper limit. 1 π [ − cos x ] 0 π = 1 π [ − cos ( π ) ] Substitute cos ( π ) = − 1 in the above equation, 1 π [ − cos x ] 0 π = 1 π [ − ( − 1 ) ] 1 π [ − cos x ] 0 π = 1 π Now evaluate the value of integral at the lower limit. 1 π [ − cos x ] 0 π = 1 π [ − cos ( 0 ) ] Substitute cos ( 0 ) = 1 in the above equation, 1 π [ − cos x ] 0 π = 1 π × 1 1 π [ − cos x ] 0 π = 1 π Subtract the value of the lower limit from the upper limit. 1 π ∫ 0 π sin x d x = [ − cos ( π ) + cos ( 0 ) ] 1 π ∫ 0 π sin x d x = 1 π + 1 π 1 π ∫ 0 π sin x d x = 2 π Thus, the average of the function is 2 π

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Chapter 16.3, Problem 53E

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To calculate: The average of the function f(x)=sinx over the interval [0,π] and to sketch the graph of the function and its average.

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Chapter 16.3, Problem 52E

Chapter 16.3, Problem 54E

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Chapter 16 Solutions

EBK FINITE MATH AND APPLIED CALCULUS

Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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The average of the function f ( x ) = sin x over the interval [ 0 , π ] and to sketch the graph of the function and its average.

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To calculate: The average of the function f(x)=sinx over the interval [0,π] and to sketch the graph of the function and its average.

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Chapter 16 Solutions