   Chapter 15.6, Problem 21E

Chapter
Section
Textbook Problem

Use a triple integral to find the volume of the given solid.21. The solid enclosed by the cylinder y = x2 and the planes z = 0 and y + z = 1

To determine

To find: The volume of the solid enclosed by the cylinder y=x2 and the planes z=0 and y+z=1 by using the triple integral.

Explanation

Let E be the region enclosed by the tetrahedron where

E={(x,y,z)|1x1,x2y1,0z1y}.

The volume of integral for the given is, EdV=11x2101ydzdydx.

Integrate the given integral with respect to z and apply the limit of it.

EdV=11x21[z]01ydydx=11x21[1y0]dydx=11x21[1y]dydx

Integrate the given integral with respect to y and apply the limit of it.

EdV=11[yy22]x21dx=11[(1122)][((x2)(x2)22)]dx=11(12x2+x42)dx

Integrate the given integral with respect to x and apply the limit of it

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