Question

, where \({\rm{T}}\) is the solid tetrahedron with vertices \({\rm{(0,0,0),(1,0,0),(1,1,0), and (1,0,1)}}\).

Short answer

The required answer is\(\frac{{\rm{1}}}{{{\rm{30}}}}\).

Step-by-step solution

Concept Introduction

Triple integrals are the three-dimensional equivalents of double integrals. They're a way to add up an unlimited number of minuscule quantities connected with points in a three-dimensional space.

If the area under the curve from to is the definite integral of a function of one variable, then the double integral is equal to the volume under the surface and above the -plane in the integration region.

Draw the structure of tetrahedron

To begin, draw the tetrahedron as follows:

Write blue plane's equation

The triangle with vertices\({\rm{(0,0,0),(1,0,0),(0,1,0)}}\) represents the solid's projection on the plane\({\rm{x y}}\). When x increases from 0 to\({\rm{1, y}}\) increases from 0 to the line that connects the coordinates\({\rm{(1,0,0)}}\) and\({\rm{(0,1,0)}}\). Determine this line's equation:

\(\begin{array}{l}\frac{{{\rm{y - 0}}}}{{{\rm{1 - 0}}}}{\rm{ = }}\frac{{{\rm{x - 1}}}}{{{\rm{0 - 1}}}}\\{\rm{y = 1 - x}}\end{array}\)

Then we have to figure out what the limits of z are, which range from the blue triangle to one. Determine the equation for the blue plane, which may be expressed as follows because it passes through the origin:

\({\rm{x + a y + b z = 0}}\)

\({\rm{Determine a, b :}}\)

\(\begin{array}{c}\left\{ {\begin{array}{*{20}{l}}{{\rm{1 + a(0) + b(1) = 0}}}\\{{\rm{0 + a(1) + b(1) = 0}}}\end{array}} \right.{\rm{ }}\\{\rm{\{ }}\begin{array}{*{20}{l}}{{\rm{1 + b = 0}}}\\{{\rm{a + b = 0}}}\end{array}\end{array}\)

\(\begin{array}{l}{\rm{b = - 1}}\\{\rm{a = 1}}\end{array}\)

The blue plane's equation is:

\(\begin{array}{l}{\rm{x + y - z = 0}}\\{\rm{z = x + y}}\end{array}\)

As a result,\({\rm{z}}\)goes from\({\rm{x + y}}\)to 1. We have the following:

\({\rm{D = \{ (x,y,z)}}\mid {\rm{0}} \le {\rm{x}} \le {\rm{1,0}} \le {\rm{y}} \le {\rm{1 - x,x + y}} \le {\rm{z}} \le {\rm{1\} }}\)

Find the value of

Assess the integral:

Therefore, the required solution is \(\frac{{\rm{1}}}{{{\rm{30}}}}\).