Explanation Given: T he equations of the two surfaces z = x 2 + y 2 and z = 8 − x 2 − y 2 Calculation: Step 1: We first must identify the domain of integration in the xy -plane. To do this we equate the equations of the two surfaces. That is, x 2 + y 2 = 8 − x 2 − y 2 ⇒ 2 ( x 2 + y 2 ) = 8 ⇒ x 2 + y 2 = 4 Therefore, the domain of integration is D = { ( x , y ) ; x 2 + y 2 ≤ 4 } Step 2: The double integrals ∬ D ( x 2 + y 2 ) d A and ∬ D ( 8 − x 2 − y 2 ) d A give the volumes of the regions bounded by the Graphs of z = x 2 + y 2 and z = 8 − x 2 − y 2 ,respectively, and the xy -plane over the region D . Therefore, the volume of the required region is the difference between these two integrals (see figure). That is, v = ∬ D ( 8 − x 2 − y 2 ) d A − ∬ D ( x 2 + y 2 ) d A = ∬ D ( 8 − x 2 − y 2 ) − ( x 2 + y 2 ) d A

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ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.4, Problem 39E

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To calculate: The volume of region in the given figure.

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Chapter 16.4, Problem 38E

Chapter 16.4, Problem 40E

Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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The volume of region in the given figure.

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To calculate: The volume of region in the given figure.

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Chapter 16 Solutions