Explanation Formula used: The left Riemann sum for the function f ( x ) defined on the interval [ a , b ] for n subinterval is ∑ k = 1 n f ( x k * ) Δ x = ∑ k = 1 n f [ a + ( k − 1 ) Δ x ] Δ x . Calculation: Divide the interval [ 0 , π ] into n equal subintervals with length Δ x = b − a n = π − 0 n = π n . Computation of area by using left Riemann sum is as follows. A n = ∑ k = 1 n f ( x k * ) Δ x = ∑ k = 1 n f [ 0 + ( k − 1 ) Δ x ] Δ x [ ∵ x k * = a + ( k − 1 ) Δ x ] = ∑ k = 1 n f [ ( k − 1 ) ( π n ) ] × π n = ∑ k = 0 n − 1 [ | cos ( 2 ⋅ k π n ) | ] × π n [ ∵ f ( x ) = | cos 2 x | ] Now, for n = 16 subintervals the area of the region is as follows. A 16 = ∑ k = 0 15 [ | cos ( 2 ⋅ k π 16 ) | ] × π 16 = [ | cos ( 0 ) | + | cos ( 2 ⋅ π 16 ) | + | cos ( 2 ⋅ 2 π 16 ) | + ⋯ + | cos ( 2 ⋅ 15 π 16 ) | ] × π 16 = 1

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 5.1, Problem 74E

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To approximate: The area of the region bounded by the function f(x)=|cos2x| and the x-axis in the interval [0,π] with n=16, 32, 64 subintervals by using the left Riemann sum.

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Chapter 5.1, Problem 73E

Chapter 5.1, Problem 75E

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To approximate: The area of the region bounded by the function f ( x ) = | cos 2 x | and the x -axis in the interval [ 0 , π ] with n = 16 , 32 , 64 subintervals by using the left Riemann sum.

Solution Summary: The author approximates the area of the region bounded by the function f(x) and the x -axis using the left Riemann sum.

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To approximate: The area of the region bounded by the function f(x)=|cos2x| and the x-axis in the interval [0,π] with n=16, 32, 64 subintervals by using the left Riemann sum.

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Chapter 5 Solutions