(a) Explanation Given: The function f ( x ) = cos ( 2 x ) defined on interval [ 0 , π 4 ] . Formula used: (1). 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 . (2). The right 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 ) Δ x ] Δ x . (3). The midpoint 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 2 ) Δ x ] Δ x . Calculation: First find the length of the each subinterval, Δ x = π 4 − 0 60 = π 240 Then, the subintervals are as follows. [ 0 , π 240 ] , [ π 240 , 2 π 240 ] , [ 2 π 240 , 3 π 240 ] , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , [ 59 π 240 , π 4 ] The left end points of the above subintervals are 0 , π 240 , 2 π 240 , 3 π 240 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 59 π 240 and the right end points are π 240 , 2 π 240 , 3 π 240 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 59 π 240 , π 4 and the mid points are π 480 , 3 π 480 , 4 π 480 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 119 π 480 . Computation of the left Riemann sum as follows. ∑ k = 1 60 f ( x k * ) Δ x = ∑ k = 1 60 f [ 0 + ( k − 1 ) Δ x ] Δ x [ ∵ x k * = a + ( k − 1 ) Δ x ] = ∑ k = 1 60 f [ 0 + ( k − 1 ) π 240 ] × π 240 = ∑ k = 1 60 cos [ ( k − 1 ) π 120 ] × π 240 [ ∵ f ( x ) = cos ( 2 x ) ] = ∑ k = 0 59 cos [ k π 120 ] × π 240 On further simplification, ∑ k = 1 60 f ( x k * ) Δ x = ∑ k = 0 59 cos [ k π 120 ] × π 240 = [ cos ( 0 ) + cos ( π 120 ) + cos ( 2 π 120 ) + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + cos ( 59 π 120 ) ] × π 240 = [ 1 + cos ( π 120 ) + cos ( 2 π 120 ) + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + cos ( 59 π 120 ) ] × π 240 = 0.507 Computation of the right Riemann sum as follows (b) To determine To estimate: The area of the region bounded by the graph of the function f ( x ) and the x -axis on the interval [ 0 , π 4 ] .

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

2nd Edition

ISBN: 9780321954237

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

Publisher: PEARSON

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Question

Chapter 5.1, Problem 46E

(a)

To determine

To write: The left Riemann sum, right Riemann sum and the midpoint Riemann sum in the form of sigma notation for n=60 .

(b)

To determine

To estimate: The area of the region bounded by the graph of the function f(x) and the x -axis on the interval [0,π4] .

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Chapter 5.1, Problem 45E

Chapter 5.1, Problem 47E

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