   Chapter 15.6, Problem 19E

Chapter
Section
Textbook Problem

Use a triple integral to find the volume of the given solid.19. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4

To determine

To find: The volume of the solid tetrahedron enclosed by the coordinate planes and the plane 2x+y+z=4 by using the triple integral.

Explanation

Let E be the region enclosed by the tetrahedron where E={(x,y,z)|0x2,0y42x,0z42xy}

The volume of integral for the given tetrahedron is, EdV=02042x042xydzdydx

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

EdV=02042x[z]042xydydx=02042x[42xy0]dydx=02042x[42xy]dydx

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

EdV=02[(42x)yy22]042xdx=02[((42x

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