Q. 71 Let聽R=x,聽y,聽z聽|聽a1猢絰猢絘2... [FREE SOLUTION]

Chapter 13: Q. 71 (page 1057)

$L e t R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\} . P r o v e t h a t : {鈭�}_{R} d V = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) . W h a t i s t h e r e l a t i o n s h i p b e t w e e n R a n d t h e p r o d u c t (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) .$

Short Answer

Expert verified

${鈭�}_{R} d V = {鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} {鈭�}_{c_{1}}^{c_{2}} d z d y d x = {鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} [{鈭�}_{c_{1}}^{c_{2}} d z] d y d x = (c_{2} - c_{1}) {鈭�}_{a_{1}}^{a_{2}} [{鈭�}_{b_{1}}^{b_{2}} d y] d x = (c_{2} - c_{1}) (b_{2} - b_{1}) {鈭�}_{a_{1}}^{a_{2}} d x = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) {鈭�}_{R} d V = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) .$

Step by step solution

Step 1. Given Information.

$G i v e n : R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\}$

Step 2. Proof.

$T h e t r i p l e i n t e g r a l i s e v a l u a t e d a s f o l l o w s : {鈭�}_{R} d V = {鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} {鈭�}_{c_{1}}^{c_{2}} d z d y d x = {鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} [{鈭�}_{c_{1}}^{c_{2}} d z] d y d x = {鈭�}_{a_{1}}^{a_{2}} {鈭�}_{b_{1}}^{b_{2}} (c_{2} - c_{1}) d y d x = (c_{2} - c_{1}) {鈭�}_{a_{1}}^{a_{2}} [{鈭�}_{b_{1}}^{b_{2}} d y] d x = (c_{2} - c_{1}) {鈭�}_{a_{1}}^{a_{2}} (b_{2} - b_{1}) d x = (c_{2} - c_{1}) (b_{2} - b_{1}) {鈭�}_{a_{1}}^{a_{2}} d x = (c_{2} - c_{1}) (b_{2} - b_{1}) (a_{2} - a_{1}) = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) = R H S . H e n c e, p r o v e d .$

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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LetR=x,y,z|a1猢�x猢�a2,b1猢�y猢�b2,c1猢�z猢�c2.Provethat:鈭�RdV=a2-a1b2-b1c2-c1.WhatistherelationshipbetweenRandtheproducta2-a1b2-b1c2-c1.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof.

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