Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Let ρ(x,y,z)be a density function defined on the rectangular solid Rwhere role="math" localid="1650360487967" R={(x,y,z)|a1xa2,b1yb2,andc1zc2}. Set up iterated integrals representing the mass of R, using all six distinct orders of integration.

Short Answer

Expert verified

The six distinct orders of integration representing the mass of Rare,

role="math" localid="1650361542302" a1a2b1b2c1c2ρx,y,zdzdydxb1b2a1a2c1c2ρx,y,zdzdxdya1a2c1c2b1b2ρx,y,zdydzdxc1c2a1a2b1b2ρx,y,zdydxdzb1b2c1c2a1a2ρx,y,zdxdzdyc1c2b1b2a1a2ρx,y,zdxdydz

Step by step solution

01

Step 1. Given information

R={(x,y,z)|a1xa2,b1yb2,andc1zc2}.

02

Step 2. The definition of the iterated triple integral:

If ρx,y,zbe a density function defined on the rectangular solid Rwhere role="math" localid="1650361021093" R={(x,y,z)|axb,cyd,andezf}then the mass of Ris defined as, Ωρx,y,zdV=abcdefρx,y,zdzdydx.

The other mass equations are defined in the same way for other 5triple integrals. Given that ρx,y,zbe a density function defined on the rectangular solid Rwhere R={(x,y,z)|a1xa2,b1yb2,andc1zc2}then the mass of Ris defined as role="math" localid="1650361522159" Ωρx,y,zdV=a1a2b1b2c1c2ρx,y,zdzdydx.

03

Step 3. The other five equations for the mass are,

b1b2a1a2c1c2ρx,y,zdzdxdya1a2c1c2b1b2ρx,y,zdydzdxc1c2a1a2b1b2ρx,y,zdydxdzb1b2c1c2a1a2ρx,y,zdxdzdyc1c2b1b2a1a2ρx,y,zdxdydz

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Study anywhere. Anytime. Across all devices.

Sign-up for free