Explanation Given information: The Equation of the plane is y = 1 − x . The Equation of the surface is z = cos ( π x / 2 ) . Calculation: From the Equations of plane and surface, Take the range of x varies from 0 to 1. Take the range of y varies from 0 to 1 − x . Take the range of z varies from 0 to cos ( π x / 2 ) . Express the given integral as follows: ∫ 0 1 ∫ 0 1 − x ∫ 0 cos ( π x / 2 ) d z d y d x (1) Integrate Equation (1) with respect to z . ∫ 0 1 ∫ 0 1 − x ∫ 0 cos ( π x / 2 ) d z d y d x = ∫ 0 1 ∫ 0 1 − x ( z ) 0 cos ( π x / 2 ) d y d x = ∫ 0 1 ∫ 0 1 − x ( cos ( π x 2 ) − 0 ) d y d x = ∫ 0 1 ∫ 0 1 − x ( cos ( π x 2 ) ) d y d x (2) Integrate Equation (2) with respect to y . ∫ 0 1 ∫ 0 1 − x ( cos ( π x 2 ) ) d y d x = ∫ 0 1 cos ( π x 2 ) ( y ) 0 1 − x d x = ∫ 0 1 cos ( π x 2 ) ( 1 − x − 0 ) d x = ∫ 0 1 cos ( π x 2 ) ( 1 − x ) d x = ∫ 0 1 cos ( π x 2 ) d x − ∫ 0 1 x cos ( π x 2 ) d x = ( sin π x 2 π 2 ) 0 1 − ∫ 0 1 x cos ( π x 2 ) d x = 2 π ( sin π x 2 ) 0 1 − ∫ 0 1 x cos ( π x 2 ) d x (3) From Equation (3), Take the value of u as ( π x 2 )

University Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780321999580

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.5, Problem 28E

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Find the volume of the octant which is bounded by the plane y=1−x and the surface z=cos(πx/2).

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Chapter 14.5, Problem 27E

Chapter 14.5, Problem 29E

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Chapter 14 Solutions

University Calculus: Early Transcendentals (3rd Edition)

Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - Prob. 10E

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Find the volume of the octant which is bounded by the plane y = 1 − x and the surface z = cos ( π x / 2 ) .

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Find the volume of the octant which is bounded by the plane y=1−x and the surface z=cos(πx/2).

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Chapter 14 Solutions