Explanation Given information: The integral is ∫ 0 2 ∫ 1 4 ∫ − 1 2 ( 8 x − 2 y + z ) d x d y d z . Formula used: Integral of power function is ∫ a x n d x = a [ x n + 1 n + 1 ] + C Calculation: Consider the function, ∫ 0 2 ∫ 1 4 ∫ − 1 2 ( 8 x − 2 y + z ) d x d y d z = ∫ 0 2 ( ∫ 1 4 ( ∫ − 1 2 ( 8 x − 2 y + z ) d x ) d y ) d z First evaluate the inner integral, Then, ∫ − 1 2 ( 8 x − 2 y + z ) d x = 8 [ x 1 + 1 1 + 1 ] − 1 2 − 2 y [ x 0 + 1 0 + 1 ] − 1 2 + z [ x 0 + 1 0 + 1 ] − 1 2 = 4 [ x 2 ] − 1 2 − 2 y [ x ] − 1 2 + z [ x ] − 1 2 Now evaluate the limits, ∫ − 1 2 ( 8 x − 2 y + z ) d x = 4 [ x 2 ] − 1 2 − 2 y [ x ] − 1 2 + z [ x ] − 1 2 = 4 [ ( 2 ) 2 − ( − 1 ) 2 ] − 2 y [ ( 2 ) − ( − 1 ) ] + z [ ( 2 ) − ( − 1 ) ] = 4 [ 4 − 1 ] − 2 y [ 2 + 1 ] + z [ 2 + 1 ] = 12 − 6 y + 3 z Now inner integral is ∫ − 1 2 ( 8 x − 2 y + z ) d x = 12 − 6 y + 3 z

Calculus and Its Applications (11th Edition)

11th Edition

ISBN: 9780321979391

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Chapter 6.6, Problem 24E

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To calculate: The value of given integral function ∫02∫14∫−12(8x−2y+z)dxdydz.

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Chapter 6.6, Problem 23E

Chapter 6.6, Problem 25E

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The value of given integral function ∫ 0 2 ∫ 1 4 ∫ − 1 2 ( 8 x − 2 y + z ) d x d y d z .

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To calculate: The value of given integral function ∫02∫14∫−12(8x−2y+z)dxdydz.

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