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LetR=x,y,z|a1xa2,b1yb2,c1zc2.Provethat:RdV=a2-a1b2-b1c2-c1.WhatistherelationshipbetweenRandtheproducta2-a1b2-b1c2-c1.

Short Answer

Expert verified

RdV=a1a2b1b2c1c2dzdydx=a1a2b1b2c1c2dzdydx=c2-c1a1a2b1b2dydx=c2-c1b2-b1a1a2dx=a2-a1b2-b1c2-c1RdV=a2-a1b2-b1c2-c1.

Step by step solution

01

Step 1. Given Information.

Given:R=x,y,z|a1xa2,b1yb2,c1zc2

02

Step 2. Proof.

Thetripleintegralisevaluatedasfollows:RdV=a1a2b1b2c1c2dzdydx=a1a2b1b2c1c2dzdydx=a1a2b1b2c2-c1dydx=c2-c1a1a2b1b2dydx=c2-c1a1a2b2-b1dx=c2-c1b2-b1a1a2dx=c2-c1b2-b1a2-a1=a2-a1b2-b1c2-c1=RHS.Hence,proved.

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