Skip to main content
HomeMATHalinoEngineering Math Review

Search form

Login • Register

  • Home
    • Recent
    • Glossary
    • About
  • Algebra
    • Derivation of Formulas
    • Engineering Economy
    • General Engineering
  • Trigo
    • Spherical Trigonometry
  • Geometry
    • Solid Geometry
    • Analytic Geometry
  • Calculus
    • Integral Calculus
    • Differential Equations
    • Advance Engineering Mathematics
  • Mechanics
    • Strength of Materials
    • Structural Analysis
  • CE
    • CE Board: Math
    • CE Board: Hydro Geo
    • CE Board: Design
    • Surveying
    • Hydraulics
    • Timber Design
    • Reinforced Concrete
    • Geotechnical Engineering
  • Courses
    • Exams
    • Old MCQ
  • Forums
    • Basic Engineering Math
    • Calculus
    • Mechanics
    • General Discussions
  • Blogs

Breadcrumbs

You are here:

  1. Home
  2. Integration of Polar Area

Integration of Polar Area

03 Area Enclosed by Cardioids: r = a(1 + sin θ); r = a(1 - sin θ), r = a(1 + cos θ), r = a(1 - cos θ)

Problem
Find the area individually enclosed by the following Cardioids:
(A)   $r = a(1 - \cos \theta)$
(B)   $r = a(1 + \cos \theta)$
(C)   $r = a(1 - \sin \theta)$
(D)   $r = a(1 + \sin \theta)$
 

003-cardioid-neg-pos-sine-cosine.gif

 

  • Read more about 03 Area Enclosed by Cardioids: r = a(1 + sin θ); r = a(1 - sin θ), r = a(1 + cos θ), r = a(1 - cos θ)
  • Log in or register to post comments

08 Area Enclosed by r = a sin 3θ and r = a cos 3θ

Problem
Find the area bounded by $r = a \sin 3\theta$ and $r = a \cos 3\theta$.
 

008-polar-area-three-leaf_rose_sine_cosine.gif

 

  • Read more about 08 Area Enclosed by r = a sin 3θ and r = a cos 3θ
  • Log in or register to post comments

05 Area Enclosed by r = a sin 2θ and r = a cos 2θ

Problem
Find the area bounded by $r = a \sin 2\theta$ and $r = a \cos 2\theta$.
 

008-polar-area-four-leaf_sine_cosine.gif

 

  • Read more about 05 Area Enclosed by r = a sin 2θ and r = a cos 2θ
  • Log in or register to post comments

06 Area Within the Curve r^2 = 16 cos θ

Example 6
What is the area within the curve r2 = 16 cos θ?
 

  • Read more about 06 Area Within the Curve r^2 = 16 cos θ
  • Log in or register to post comments

05 Area Enclosed by Four-Leaved Rose r = a cos 2θ

Find the area enclosed by four-leaved rose r = a cos 2θ.
 

  • Read more about 05 Area Enclosed by Four-Leaved Rose r = a cos 2θ
  • Log in or register to post comments

04 Area of the Inner Loop of the Limacon r = a(1 + 2 cos θ)

Example 4
Find the area of the inner loop of the limacon r = a(1 + 2 cos θ).
 

  • Read more about 04 Area of the Inner Loop of the Limacon r = a(1 + 2 cos θ)
  • Log in or register to post comments

03 Area Inside the Cardioid r = a(1 + cos θ) but Outside the Circle r = a

Example 3
Find the area inside the cardioid r = a(1 + cos θ) but outside the circle r = a.
 

  • Read more about 03 Area Inside the Cardioid r = a(1 + cos θ) but Outside the Circle r = a
  • Log in or register to post comments

02 Area Bounded by the Lemniscate of Bernoulli r^2 = a^2 cos 2θ

Example 2
Find the area bounded by the lemniscate of Bernoulli r2 = a2 cos 2θ.
 

  • Read more about 02 Area Bounded by the Lemniscate of Bernoulli r^2 = a^2 cos 2θ
  • Log in or register to post comments

Plane Areas in Polar Coordinates | Applications of Integration

The fundamental equation for finding the area enclosed by a curve whose equation is in polar coordinates is...
 

$\displaystyle A = \frac{1}{2}{\int_{\theta_1}^{\theta_2}} r^2 \, d\theta$

 

  • Read more about Plane Areas in Polar Coordinates | Applications of Integration
  • Log in or register to post comments
Home • Forums • Blogs • Glossary • Recent
About • Contact us • Terms of Use • Privacy Policy • Hosted by Linode • Powered by Drupal
MATHalino - Engineering Mathematics • Copyright 2025 Jhun Vert • All rights reserved