Essential Concepts
- The area of a region in polar coordinates defined by the equation r=f(θ)r=f(θ) with α≤θ≤βα≤θ≤β is given by the integral A=12∫βα[f(θ)]2dθA=12∫βα[f(θ)]2dθ.
- To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.
- The arc length of a polar curve defined by the equation r=f(θ)r=f(θ) with α≤θ≤βα≤θ≤β is given by the integral L=∫βα√[f(θ)]2+[f′(θ)]2dθ=∫βα√r2+(drdθ)2dθ.
Key Equations
- Area of a region bounded by a polar curve
A=12∫βα[f(θ)]2dθ=12∫βαr2dθ - Arc length of a polar curve
L=∫βα√[f(θ)]2+[f′(θ)]2dθ=∫βα√r2+(drdθ)2dθ
Candela Citations
CC licensed content, Shared previously
- Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-3/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction