Answered: Find the average x-coordinate of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find the average x-coordinate of the points in the prism
D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 3 - 3x, 0 ≤ z ≤ 2}.

Expert Solution

Step 1: Objective

To find: The average x-coordinate of the points in the prism $D space equals open curly brackets left parenthesis x comma y comma z right parenthesis colon 0 less or equal than space x space less or equal than 1 comma 0 less or equal than space y space less or equal than 3 minus 3 x comma 0 less or equal than space z space less or equal than 2 close curly brackets$ .

Step 2: Finding the average x-coordinate of the points in the prism

Volume of the prism is given by

$table attributes columnalign right center left columnspacing 0px end attributes row V equals cell integral subscript 0 superscript 1 integral subscript 0 superscript 3 minus 3 x end superscript integral subscript 0 superscript 2 1 d z d y d x end cell row blank equals cell integral subscript 0 superscript 1 integral subscript 0 superscript 3 minus 3 x end superscript open parentheses 2 close parentheses d y d x end cell row blank equals cell 2 integral subscript 0 superscript 1 open vertical bar y close vertical bar subscript 0 superscript 3 minus 3 x end superscript d x end cell row blank equals cell 2 integral subscript 0 superscript 1 3 minus 3 x d x end cell row blank equals cell 2 open vertical bar 3 x minus fraction numerator 3 x squared over denominator 2 end fraction close vertical bar subscript 0 superscript 1 end cell row blank equals cell 2 open parentheses 3 minus 3 over 2 close parentheses end cell row blank equals cell 2 cross times 3 over 2 end cell row blank equals 3 end table$

Hence, $V equals space 3$

The average x-coordinate of the points in the prism is given by :

$table attributes columnalign right center left columnspacing 0px end attributes row cell top enclose x end cell equals cell 1 over V integral subscript 0 superscript 1 integral subscript 0 superscript 3 minus 3 x end superscript integral subscript 0 superscript 2 x d z d y d x end cell row blank equals cell 1 third integral subscript 0 superscript 1 integral subscript 0 superscript 3 minus 3 x end superscript x open vertical bar z close vertical bar subscript 0 superscript 2 d y d x end cell row blank equals cell 2 over 3 integral subscript 0 superscript 1 integral subscript 0 superscript 3 minus 3 x end superscript x d y d x end cell row blank equals cell 2 over 3 integral subscript 0 superscript 1 x open vertical bar y close vertical bar subscript 0 superscript 3 minus 3 x end superscript d x end cell row blank equals cell 2 over 3 integral subscript 0 superscript 1 x open parentheses 3 minus 3 x close parentheses d x end cell row blank equals cell 2 over 3 open vertical bar fraction numerator 3 x squared over denominator 2 end fraction minus fraction numerator 3 x cubed over denominator 3 end fraction close vertical bar subscript 0 superscript 1 end cell row blank equals cell 2 over 3 open parentheses 3 over 2 minus 1 close parentheses end cell row blank equals cell 2 over 3 cross times 1 half end cell row blank equals cell 1 third end cell end table$

Hence, the average x-coordinate of the points in the prism is $1 third.$

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