Explanation Given information: The function is given by f ( x ) = 4 − x , [ 0 , 6 ] and n = 4 . Calculation: Consider the graph of f ( x ) = 4 − x . By the graph, Area = Area of triangle ABC − Area of triangle CDE = ( 1 2 × 4 × 4 ) − ( 1 2 × 2 × 2 ) = 6 To find left, right, midpoint, upper, lower, trapezoid sums, Let’s take Δ x = b − a n = 6 − 0 4 = 1.5 and x k = 0 + k Δ x = 1.5 k The 4-rectangle left sum for f on [0,6] is ∑ k = 1 4 f ( x k − 1 ) Δ x = ∑ k = 1 4 f ( 1.5 ( k − 1 ) ) Δ x = ∑ k = 1 4 f ( 1.5 k − 1.5 ) Δ x = f ( 0 ) Δ x + f ( 1.5 ) Δ x + f ( 3 ) Δ x + f ( 4.5 ) Δ x = ( 4 × 1.5 ) + ( 2.5 × 1.5 ) + ( 1 × 1.5 ) + ( ( − 0.5 ) × 1.5 ) = 12.75 The 4-rectangle right sum for f on [0,6] is ∑ k = 1 4 f ( x k ) Δ x = ∑ k = 1 4 f ( 1.5 k ) Δ x = f ( 1.5 ) Δ x + f ( 3 ) Δ x + f ( 4.5 ) Δ x + f ( 6 ) Δ x = ( 2.5 × 1.5 ) + ( 1 × 1.5 ) + ( ( − 0.5 ) × 1.5 ) + ( ( − 2 ) × 1.5 ) = 1.5 The 4-rectangle mid point sum for f on [0,6] is ∑ k = 1 4 f ( x k − 1 + x k 2 ) Δ x = ∑ k = 1 4 f ( ( 1.5 ( k − 1 ) ) + ( 1.5 k ) 2 ) Δ x = ∑ k = 1 4 f ( 3 k − 1.5 2 ) Δ x = f ( 0.75 ) Δ x + f ( 2.25 ) Δ x + f ( 3.75 ) Δ x + f ( 5.25 ) Δ x = ( 3

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ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 4.2, Problem 36S

To determine

To calculate: Calculate the area between f and x-axis using the area of circle, triangle and rectangle. Then calculate exact area and left, right, midpoint, upper, lower, trapezoid sums.

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Chapter 4.2, Problem 35S

Chapter 4.2, Problem 37S

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Calculus

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Calculate the area between f and x-axis using the area of circle, triangle and rectangle. Then calculate exact area and left, right, midpoint, upper, lower, trapezoid sums.

Solution Summary: The author calculates the area between f and x-axis using circle, triangle and rectangle. They then calculate exact area and left, right, midpoint, upper, lower, trapezoid sums.

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To calculate: Calculate the area between f and x-axis using the area of circle, triangle and rectangle. Then calculate exact area and left, right, midpoint, upper, lower, trapezoid sums.

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