Explanation Given: Region is bounded by z = 2 − y 2 , z = y , x = 0 , y = 0 and x + y = 1 . Formula used: The volume of a region w is V = ∫ ∫ ∫ w l d V and, the integral formula ∫ x n = x n + 1 n + 1 Calculation: First, to calculate the volume we will find the region w, as given ‘z’ varies from z = y to z = 2 − y 2 (as the region is z-simple), thus V = ∫ ∫ ∫ w d V = ∫ ∫ D ( ∫ y 2 − y 2 d z ) d A Now, to find D if we simplify the integrate in order d x d y then ‘ y’ varies from y = 0 to y = 1 − x and, ‘ x’ varies from x = 0 to x = 1 . Thus, V = ∫ 0 1 ∫ 0 1 − x ∫ y 2 − y 2 d z d y d x Integrating the inner integral with respect to ‘z’, then middle integral with respect to ‘ y’ and lastly the outer most with respect to ‘ x’, V = ∫ 0 1 ∫ 0 1 − x ∫ y 2 − y 2 d z d y d x = ∫ 0 1 ∫ 0 1 − x [ z ] y 2 − y 2 d y d x = ∫ 0 1 ∫ 0 1 − x ( 2 − y 2 − y ) d y d x = ∫ 0 1 [ 2 y

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Author: Rogawski

Publisher: MAC HIGHER

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Chapter 15.2, Problem 50E

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Compute the volume of the region bounded by z=2−y2,z=y,x=0,y=0 and x+y=1.

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Chapter 15.2, Problem 49E

Chapter 15.2, Problem 51E

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Chapter 15 Solutions

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Compute the volume of the region bounded by z = 2 − y 2 , z = y , x = 0 , y = 0 and x + y = 1 .

Solution Summary: The author explains how to calculate the volume of a region bounded by z=2-y2,z =y,x=0,y =0 and x+y=1.

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Compute the volume of the region bounded by z=2−y2,z=y,x=0,y=0 and x+y=1.

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Chapter 15 Solutions