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

Give a geometric explanation why n02π/n0Rrdrdθ=πR2

for any positive real number and any positive integer n.

Would the equation also hold for nonintegral values of n?

Short Answer

Expert verified

Area of nsectors = n02π/n0πrdrdθ=πR2[Area of a circle] is shown.

Step by step solution

01

Given Information

The objective of this problem is to give the geometric explanation why

n02π/n0πrdrdθ=πR2

02

Calculation

Suppose a circle of radius Ris divided into nsectors of equal areas. Each sector will subtend an angle of 2πn at the center of circle. Area of a sector can be calculated in polar form as an integral.

Area of a sector =02πi=0Rrdrdθ

Integrate with respect to r.

02π/n0πrdrdθ=02π/nr220πdθ

02π'/n0πrdrdθ=02sinR22dθ

02π/n0Rrdrdθ=R2202π/ndθ

Integrate with respect to θ.

02π/n0nrdrdθ=R22[θ]02π/m

Substitute the limits

02π/n0πrdrdθ=πR2n

Therefore, area ofnsectors

n02sin0Rrdrdθ=πnR2n

n02π/n0πrdrdθ=πR2[Area of a circle]

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