Explanation Given: g ( x ) = sin x in the interval [ 0 , π ] and the number of rectangles to be used for calculation of the area is 6 . Formula used: Using sigma notation, the sum of n terms a 1 , a 2 , a 3 , … , a n is written as: ∑ i = 1 n a i = a 1 + a 2 + a 3 + … + a n Calculation: The right endpoints of the six rectangles are: 0 + π 6 i = π 6 i Where, i = 1 , 2 , 3 , 4 , 5 , 6 . The width of each rectangle is: π − 0 6 = π 6 Height of each rectangle can be obtained by evaluation of values of the provided function at the right endpoints of each rectangle. Approximate area between the provided function and the x -axis is equal to the sum of the sixrectangles and the sum of the areas of the six rectangles is: Area = ∑ i = 1 6 g ( π 6 i ) ( π 6 ) Put value of g ( π 6 i ) , ∑ i = 1 6 g ( π 6 i ) ( π 6 ) = ∑ i = 1 6 [ sin ( π 6 i ) ] ( π 6 ) Factor out π 6 from the summation, ∑ i = 1 6 [ sin ( π 6 i ) ] ( π 6 ) = π 6 ∑ i = 1 6 [ sin ( π 6 i ) ] Apply the formula, ∑ i = 1 n a i = a 1 + a 2 + … + a n So, π 6 ∑ i = 1 6 [ sin ( π 6 i ) ] = π 6 ( sin ( π 6 ) + sin ( π 3 ) + sin ( π 2 ) + sin ( 2 π 3 ) + sin ( 5 π 6 ) + sin ( π ) ) = π 6 ( 0

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Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 4.2, Problem 34E

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To calculate: Approximate area of the region between the function g(x)=sinx in the interval [0,π] and x-axis.

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Chapter 4.2, Problem 33E

Chapter 4.2, Problem 35E

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Approximate area of the region between the function g ( x ) = sin x in the interval [ 0 , π ] and x -axis.

Solution Summary: The author calculates the approximate area of the region between the function g(x)=mathrmsinx and the number of rectangles to be used for calculation.

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To calculate: Approximate area of the region between the function g(x)=sinx in the interval [0,π] and x-axis.

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Chapter 4 Solutions