Explanation Given information: The function is given by f ( x ) = 1 − x 2 , [ − 1 , 1 ] and n = 4 . Calculation: Consider the graph of f ( x ) = 1 − x 2 , By the graph, Area=The are of ABC half circle = 1 2 π r 2 = 1 2 × π × 1 2 = π 2 = 1.570796326794897 = 1.6 To find left, right, midpoint, upper, lower, trapezoid sums, Let’s take Δ x = b − a n = 1 − ( − 1 ) 4 = 0.5 and x k = − 1 + k Δ x = − 1 + 0.5 k The 4-rectangle left sum for f on [ − 1 , 1 ] is ∑ k = 1 4 f ( x k − 1 ) Δ x = ∑ k = 1 4 f ( − 1 + 0.5 ( k − 1 ) ) Δ x = ∑ k = 1 4 f ( 0.5 k − 1.5 ) Δ x = f ( − 1 ) Δ x + f ( − 0.5 ) Δ x + f ( 0 ) Δ x + f ( 0.5 ) Δ x = ( 0 × 0.5 ) + ( 3 2 × 0.5 ) + ( 1 × 0.5 ) + ( 3 2 × 0.5 ) = 3 2 + 0.5 = 1.3660 The 4-rectangle right sum for f on [ − 1 , 1 ] is ∑ k = 1 4 f ( x k ) Δ x = ∑ k = 1 4 f ( − 1 + 0.5 k ) Δ x = f ( − 0.5 ) Δ x + f ( 0 ) Δ x + f ( 0.5 ) Δ x + f ( 1 ) Δ x = ( 3 2 × 0.5 ) + ( 1 × 0.5 ) + ( 3 2 × 0.5 ) + ( 0 × 0.5 ) = 3 2 + 0.5 = 1.3660 The 4-rectangle mid point sum for f on [ − 1 , 1 ] is ∑ k = 1 4 f ( x k − 1 + x k 2 ) Δ x = ∑ k = 1 4 f ( ( − 1 + 0.5 ( k − 1 ) ) + ( − 1 + 0.5 k ) 2 ) Δ x = ∑ k = 1 4 f ( k − 2.5 2 ) Δ x = f ( − 0.75 ) Δ x + f ( − 0.25 ) Δ x + f ( 0.25 ) Δ x + f ( 0.75 ) Δ x = ( 7 4 × 0

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Question

Chapter 4.2, Problem 37S

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 36S

Chapter 4.2, Problem 38S

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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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Chapter 4 Solutions