Explanation Given information: The function is given by f ( x ) = 3 + 4 − x 2 , [ − 2 , 2 ] and n = 4 . Calculation: Consider the graph of f ( x ) = 3 + 4 − x 2 , By the graph, Area=The area of ABDE rectangle+The area of BCD half circle = 3 × 4 + ( 1 2 × π r 2 ) = 12 + ( 1 2 × π × 2 2 ) = 12 + 2 π = 18.3 To find left, right, midpoint, upper, lower, trapezoid sums, Let’s take Δ x = b − a n = 2 − ( − 2 ) 4 = 1 and x k = − 2 + k Δ x = − 2 + k The 4-rectangle left sum for f on [-2,2] is ∑ k = 1 4 f ( x k − 1 ) Δ x = ∑ k = 1 4 f ( − 2 + k − 1 ) Δ x = ∑ k = 1 4 f ( k − 3 ) Δ x = f ( − 2 ) Δ x + f ( − 1 ) Δ x + f ( 0 ) Δ x + f ( 1 ) Δ x = 3 × 1 + ( 3 + 3 ) × 1 + 5 × 1 + ( 3 + 3 ) × 1 = 2 3 + 14 = 17.46 The 4-rectangle right sum for f on [-2,2] is ∑ k = 1 4 f ( x k ) Δ x = ∑ k = 1 4 f ( − 2 + k ) Δ x = ∑ k = 1 4 f ( k − 2 ) Δ x = f ( − 1 ) Δ x + f ( 0 ) Δ x + f ( 1 ) Δ x + f ( 2 ) Δ x = ( 3 + 3 ) × 1 + 5 × 1 + ( 3 + 3 ) × 1 + 3 × 1 = 2 3 + 14 = 17.46 The 4-rectangle mid point sum for f on [-2,2] is ∑ k = 1 4 f ( x k − 1 + x k 2 ) Δ x = ∑ k = 1 4 f ( ( − 2 + k − 1 ) + ( − 2 + k ) 2 ) Δ x = ∑ k = 1 4 f ( 2 k − 5 2 ) Δ x = f ( − 1.5 ) Δ x + f ( − 0.5 ) Δ x + f ( 0.5 ) Δ x + f ( 1.5 ) Δ x = ( 6 + 7 2 ) × 1 + ( 6 + 15 2 ) × 1 + ( 6 + 15 2 ) × 1 + ( 6 + 7 2 ) × 1 = 15 + 7 + 12 = 18

14th Edition

ISBN: 9781464140051

Author: TAALMAN, Laura, Kohn, Peter

Publisher: W. H. Freeman

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Chapter 4.2, Problem 38S

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 37S

Chapter 4.2, Problem 39S

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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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