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M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesLearning module LM 10.1: Parametrized Curves:Learning module LM 10.2: Calculus with Parametrized Curves:Learning module LM 10.3: Polar Coordinates:Learning module LM 10.4: Areas and Lengths of Polar Curves:Area inside a polar curveArea between polar curves Arc lengths of polar curves Learning module LM 10.5: Conic Sections:Learning module LM 10.6: Conic Sections in Polar Coordinates:Chapter 12: Vectors and the Geometry of SpaceChapter 13: Vector FunctionsChapter 14: Partial DerivativesChapter 15: Multiple Integrals |
Area between polar curvesTo get the area between the polar curve $r=f(\theta)$ and the polar curve $r=g(\theta)$, we just subtract the area inside the inner curve from the area inside the outer curve. If $f(\theta) \ge g(\theta)$, this means $$\frac{1}{2}\int_a^b f(\theta)^2 - g(\theta)^2 d\theta.$$Note that this is NOT $\frac{1}{2}\int_a^b [f(\theta)-g(\theta)]^2 d\theta$!! You first square and then subtract, not the other way around. As with most ``area between two curves'' problems, the tricky thing is figuring out the beginning and ending angles. This is typically where $f(\theta)=g(\theta)$. |