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

Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.

โˆซ-50โˆซ-25-x2034+x2+y23dydx

Short Answer

Expert verified

โˆซ-50โˆซ-25-x2034+x2+y23dydx=2475107648ฯ€

Step by step solution

01

Draw the region

From the limits of integration, the region is shown below,

02

Convert into polar form

By using the following substitution,

x=rcosฮธy=rsinฮธx2+y2=r2dxdy=rdrdฮธ

The equivalent polar integral of the given integral is,

โˆซ-50โˆซ-25-x2034+x2+y23dydxโ‡’โˆซฯ€3ฯ€2โˆซ0-53(4+r2)3rdrdฮธ

03

Calculate the volume

">V=โˆซฯ€3ฯ€2โˆซ0-53(4+r2)3rdrdฮธV=โˆซฯ€3ฯ€2dฮธโˆซ0-53(4+r2)3rdrV=3ฯ€2-ฯ€โˆซ0-53(4+r2)3rdrV=ฯ€2โˆซ0-53(4+r2)3rdrSubstitute,4+r2=t2rdr=dtdr=12rdtWhenr=0,t=4Whenr=-5,t=4+(-5)2=29V=ฯ€232โˆซ4291t3dtV=3ฯ€4t-3+1-3+1429V=-3ฯ€81t2429V=-3ฯ€81841-116V=-3ฯ€816-84113456V=-3ฯ€8-82513456V=2475107648ฯ€cubicunits

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