   Chapter 15, Problem 30RE

Chapter
Section
Textbook Problem

Calculate the value of the multiple integral.30. ∭ T x y   d V , where T is the solid tetrahedron with vertices (0, 0,0), ( 1 3 , 0, 0), (0,1, 0), and (0, 0, 1)

To determine

To calculate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=xy .

The region T is the tetrahedron with vertices (0,0,0),(13,0,0),(0,1,0),(0,0,1) .

Calculation:

From the given conditions, it is observed that the equations of the planes present in the sides of the tetrahedron is given by, 3x+y=1 and 3x+y+z=1 . Therefore, x varies from 0 to 13 , y varies from 0 to 13x and z varies from 0 to 13xy . First compute the integral with respect to z and apply the limit.

ExydV=013013x013xyxydzdydx=013013x[xyz]013xydydx=013013x[xy(13xy0)]dydx=013013x(xy3x2yxy2)dydx

=013013x((x3x2)yxy2)dydx

Compute the integral with respect to y and apply the limit.

ExydV=013[(x3x2)y22xy33]013xdx=013[((x3x2)(13x)22x(13x)33)((x3x2)(0)22x(0)33)]dx=013[((x3x2)(16x+9x2)2x(19x+27x227x3)3</

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