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Area of polar curves
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Area of polar curves
In this section, we will learn how to find the area of polar curves. For polar curves, we do not really find the area under the curve, but rather the area of where the angle covers in the curve. Note that not only can we find the area of one polar equation, but we can also find the area between two polar equations. It is important to always draw the curves out so that you can locate the area you are integrating, and write the integral correctly. We will first examine a generalized formula in finding areas of polar curves. Then we will apply this to finding areas with one polar curve and then two polar curves. Make sure you know your trigonometric identities very well before tackling these questions.
Basic Concepts: Fundamental theorem of calculus, Integration using trigonometric identities, Polar coordinates
Related Concepts: Areas between curves
Lessons
To find the area of a single polar equation, we use the
following formula:
A=∫αβ21r2dθ
where α is the starting angle and β is the ending angle.
To find the area that is enclosed by two polar equations like in the picture below, we use the formula:
A=∫αβ21(r2outer−r2inner)dθ
where router is the outer part of the first polar equation, and rinner is the inner part of the second polar equation.
A=∫αβ21r2dθ
where α is the starting angle and β is the ending angle.
To find the area that is enclosed by two polar equations like in the picture below, we use the formula:
A=∫αβ21(r2outer−r2inner)dθ
where router is the outer part of the first polar equation, and rinner is the inner part of the second polar equation.

- IntroductionArea of Polar Curve Overview
- 1.Area Enclosed with One Polar Equation
Find the area of the inner loop of r=1+2sinθ. - 2.Find the area of the inner loop of r=1−2cosθ.
- 3.Area Enclosed by two Polar Equations
Find the Area enclosed outside of r=4 and inside r=9. - 4.Find the Area enclosed outside r=1 and inside r=1+sinθ.
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6.
Parametric Equations and Polar Coordinates
6.1
Defining curves with parametric equations
6.2
Tangent and concavity of parametric equations
6.3
Area of parametric equations
6.4
Arc length and surface area of parametric equations
6.5
Polar coordinates
6.6
Tangents of polar curves
6.7
Area of polar curves
6.8
Arc length of polar curves