Explanation Definition used: Regular y region: “A region R in the xy plane is a regular y region if there exist functions h ( y ) and k ( y ) , the numbers c and d such that R = { ( x , y ) | h ( y ) ≤ x ≤ k ( y ) , c ≤ y ≤ d } ”. Calculation: Use the online graphing calculator to sketch the graph of y = 1 2 x , y = 6 − x , y = 1 as shown in below Figure 1. From Figure 1, it is observed that the graph for the given equations represents the regular y region. The region can be defined as R = { ( x , y ) | 1 ≤ y ≤ 2 , 2 y ≤ x ≤ 6 − y } . Evaluate the iterated integral ∬ R 1 x + y d A . Let u = x + y , then the derivative of u is d u = d x . Initially compute the integral with respect to u . ∫ 1 2 ∫ 2 y 6 − y 1 x + y d x d y = ∫ 1 2 [ ∫ 2 y 6 − y 1 u d u ] d y = ∫ 1 2 [ ln | u | ] 2 y 6 − y d y = ∫ 1 2 [ ln | x + y | ] 2 y 6 − y d y ( ∵ u = x + y ) Apply the limit value for x . ∫ 1 2 ∫ 2 y 6 − y 1 x + y d x d y = ∫ 1 2 [ ln | 6 − y + y | − ln | 2 y + y | ] d y = ∫ 1 2 [ ln 6 − ln | 3 y | ] d y = ∫ 1 2 [ ln 6 3 | y | ] d y = ∫ 1 2 [ ln 2 y ] d y Evaluate ∫ ln 2 y d v by letting u = ln ( 2 y ) and d v = d y

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 7.7, Problem 32E

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To sketch: The graph of the given equation; and to describe the region R; then to evaluate value of the iterated integral ∬R1x+ydA.

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Chapter 7.7, Problem 31E

Chapter 7.7, Problem 33E

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Chapter 7 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 7.1 - Prob. 1MP Ch. 7.1 - Matched Problem 2 Find f(2, 2, 3) for f in Example...Ch. 7.1 - Prob. 3MP Ch. 7.1 - Matched Problem 4 For the box in Example 4, find...Ch. 7.1 - Matched Problem 5 Refer to Example 5. Find the...Ch. 7.1 - Matched Problem 6 Find the coordinates of the...Ch. 7.1 - Matched Problem 7 (A) Describe the cross sections...Ch. 7.1 - Explore and Discuss 1 Imagine that you are facing...Ch. 7.1 - In Problems 18, find the indicated value of the...Ch. 7.1 - Prob. 2E

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The graph of the given equation; and to describe the region R ; then to evaluate value of the iterated integral ∬ R 1 x + y d A .

Solution Summary: The author explains how to sketch the graph of the given equation, and evaluate the iterated integral.

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To sketch: The graph of the given equation; and to describe the region R; then to evaluate value of the iterated integral ∬R1x+ydA.

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Chapter 7 Solutions