Explanation Given: The provided integral is: ∫ 0 2 ∫ y 2 3 − y d x d y Graph: Inner limit is given as y 2 ≤ x ≤ 3 − y , it means region R is bounded on the left by the line x = y 2 and on the right by line x = 3 − y . Outer limit is given as 0 ≤ y ≤ 2 , it means region R is bounded below by the x -axisand above by the line y = 2 . The equation x = y 2 is a straight line passing through origin. To draw the line x = 3 − y : y x = 3 − y − 2 5 0 3 2 1 3 0 And, to draw the line x = y 2 : y x = y 2 − 1 − 0.5 0 0 2 1 3 1.5 Therefore, the graph obtained is: Area is given by the shaded region of the graph. Find area without changing the order of the integral: ∫ 0 2 ∫ y 2 3 − y d x d y = ∫ 0 2 [ x ] y 2 3 − y d y = ∫ 0 2 ( 3 − y − y 2 ) d y = ∫ 0 2 ( 3 − 3 y 2 ) d y = [ 3 y − 3 y 2 4 ] 0 2 Further solved ahead as: ∫ 0 2 ∫ y 2 3 − y d x d y = ( 3 ⋅ 2 ) − ( 3 ⋅ 2 2 4 ) = 6 − 3 = 3 Now, change the order of integral: Divide the region into two parts as region (1) and region (2) and calculate for both parts then add them

11th Edition

ISBN: 9780357246412

Author: Ron Larson; Bruce H. Edwards

Publisher: Cengage Limited

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Chapter 14, Problem 13RE

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To graph: The region R, whose area is given by the iterated integral ∫02∫y23−ydxdy.

Also, change the order of integration and show that both orders yield the same area.

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Chapter 14, Problem 12RE

Chapter 14, Problem 14RE

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Chapter 14 Solutions

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To graph: The region R, whose area is given by the iterated integral ∫ 0 2 ∫ y 2 3 − y d x d y . Also, change the order of integration and show that both orders yield the same area.

Solution Summary: The author explains how to graph the region R, whose area is given by the iterated integral, by changing the order of integration and showing that both orders yield the same area.

Videos

To graph: The region R, whose area is given by the iterated integral ∫02∫y23−ydxdy.

Also, change the order of integration and show that both orders yield the same area.

Want to see the full answer?

Chapter 14 Solutions

To graph: The region R, whose area is given by the iterated integral ∫ 0 2 ∫ y 2 3 − y d x d y . Also, change the order of integration and show that both orders yield the same area.

Solution Summary: The author explains how to graph the region R, whose area is given by the iterated integral, by changing the order of integration and showing that both orders yield the same area.

Videos

To graph: The region R, whose area is given by the iterated integral ∫02∫y23−ydxdy. Also, change the order of integration and show that both orders yield the same area.

Want to see the full answer?

Chapter 14 Solutions

To graph: The region R, whose area is given by the iterated integral ∫02∫y23−ydxdy.

Also, change the order of integration and show that both orders yield the same area.