Explanation Given: Given integral is ∬ D y sin ( x y ) sin 2 ( π y ) d A where D is the region over the rectangle: 0 ≤ x ≤ π , 0 ≤ y ≤ 1 2 Concept used: When a double integral of constant limits is given, the integral can be evaluated by taking any order of integration . Calculation: Given integral can be evaluated easily integrating x variable first. Limits of x : 0 to π Limits of y : 0 to 1 2 ⇒ ∫ 0 1 2 ∫ 0 π y sin ( x y ) sin 2 ( π y ) d x d y ⇒ ∫ 0 1 2 ( ∫ 0 π y sin ( x y ) sin 2 ( π y ) d x ) d y ⇒ ∫ 0 1 2 ( y sin 2 ( π y ) ∫ 0 π sin ( x y ) d x ) d y ⇒ ∫ 0 1 2 ( y sin 2 ( π y ) [ − cos ( x y ) y ] 0 π ) d y ⇒ ∫ 0 1 2 ( − sin 2 ( π y ) [ cos ( x y ) ] 0 π ) d y ⇒ ∫ 0 1 2 ( − sin 2 ( π y ) [ cos ( π y ) − cos ( 0 × y ) ] ) d y ⇒ ∫ 0 1 2 ( − sin 2 ( π y ) [ cos ( π y ) − cos ( 0 ) ] ) d y ⇒ ∫ 0 1 2 ( − cos ( π y ) sin 2 ( π y ) + sin 2 ( π y ) ) d y ⇒ − ∫ 0 1 2 cos ( π y ) sin 2 ( π y ) d y + ∫ 0 1 2 sin 2 ( π y )

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Question

Chapter 12.1, Problem 60PS

To determine

To evaluate: ∬Dysin(xy)sin2(πy) dA

Where D is the region over the rectangle:

0≤x≤π,0≤y≤12

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Chapter 12.1, Problem 59PS

Chapter 12.2, Problem 1PS

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∬ D y sin ( x y ) sin 2 ( π y ) d A Where D is the region over the rectangle: 0 ≤ x ≤ π , 0 ≤ y ≤ 1 2

Solution Summary: The author explains how a double integral of constant limits can be evaluated by taking any order of integration.

Concept explainers

Videos

To evaluate: ∬Dysin(xy)sin2(πy) dA

Where D is the region over the rectangle:

0≤x≤π,0≤y≤12

Want to see the full answer?

Chapter 12 Solutions

∬ D y sin ( x y ) sin 2 ( π y ) d A Where D is the region over the rectangle: 0 ≤ x ≤ π , 0 ≤ y ≤ 1 2

Solution Summary: The author explains how a double integral of constant limits can be evaluated by taking any order of integration.

Concept explainers

Videos

To evaluate: ∬Dysin(xy)sin2(πy) dA Where D is the region over the rectangle: 0≤x≤π,0≤y≤12

Want to see the full answer?

Chapter 12 Solutions

To evaluate: ∬Dysin(xy)sin2(πy) dA

Where D is the region over the rectangle:

0≤x≤π,0≤y≤12