Explanation Definitions used: Regular x region: “A region R in the xy plane is a regular x region if there exist functions f ( x ) and g ( x ) and numbers a and b such that R = { ( x , y ) | g ( x ) ≤ y ≤ f ( x ) , a ≤ x ≤ b } ”. Regular y region: “A region R in the xy plane is a regular y region if there exist functions h ( y ) and k ( y ) and numbers c and d such that R = { ( x , y ) | h ( y ) ≤ x ≤ k ( y ) , c ≤ y ≤ d } ”. Calculation: Compute the given integral with respect to y . ∫ 0 1 [ ∫ 0 1 − x 2 x y d y ] d x = ∫ 0 1 x [ ∫ 0 1 − x 2 y d y ] d x = ∫ 0 1 x [ y 3 2 ( 3 2 ) ] 0 1 − x 2 d x = ∫ 0 1 2 x [ y 3 2 3 ] 0 1 − x 2 d x Apply the limits of y . ∫ 0 1 [ ∫ 0 1 − x 2 x y d y ] d x = ∫ 0 1 2 x [ ( 1 − x 2 ) 3 2 3 − 0 ] d x = ∫ 0 1 2 x ( 1 − x 2 ) 3 2 3 d x Use the substitution method by letting u = 1 − x 2 , then its derivative is d u = − 2 x d x d v = − d x and also the limit of v varies from 1 to 0 when x = 0 and x = 1 , respectively. That is, the above integral becomes ∫ 0 1 2 x ( 1 − x 2 ) 3 2 3 d x = ∫ 1 0 ( u ) 3 2 3 ( − d u ) = − 1 3 ∫ 1 0 u 3 2 d u Compute the integral with respect to u . − 1 3 ∫ 1 0 u 3 2 d u = − 1 3 [ u 5 2 5 2 ] 1 0 = − 2 15 [ u 5 2 ] 1 0 Apply the limit value of u . − 1 3 ∫ 1 0 u 3 2 d u = − 2 15 [ 0 − ( 1 ) 5 2 ] = − 2 15 [ − 1 ] = 2 15 Thus, the value of the given iterated integral is 2 15 . Use online graphing calculator to sketch the graph of 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 − x 2 as shown in below Figure 1. From Figure 1, it is observed that the shaded region is R . From the definition of regular x region it is clear that, the below defined order of integration indicates that the region is regular x region. That is, R = { ( x , y ) | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 − x 2 }

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14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…

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Chapter 14.7, Problem 35E

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To evaluate: The integral ∫01[∫01−x2xydy]dx and graph the region of integration, then reverse the order of the integration and do the same.

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Chapter 14.7, Problem 34E

Chapter 14.7, Problem 36E

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Chapter 14 Solutions

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

Ch. 14.1 - Matched Problem 1 Find C(20, 10) for the cost...Ch. 14.1 - Prob. 2MP Ch. 14.1 - Prob. 3MP Ch. 14.1 - Prob. 4MP Ch. 14.1 - Prob. 5MP Ch. 14.1 - Prob. 6MP Ch. 14.1 - Prob. 7MP Ch. 14.1 - Prob. 1ED Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E

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The integral ∫ 0 1 [ ∫ 0 1 − x 2 x y d y ] d x and graph the region of integration , then reverse the order of the integration and do the same.

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To evaluate: The integral ∫01[∫01−x2xydy]dx and graph the region of integration, then reverse the order of the integration and do the same.

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Chapter 14 Solutions