Summary of Area and Arc Length in Polar Coordinates

Essential Concepts

  • The area of a region in polar coordinates defined by the equation r=f(θ)r=f(θ) with αθβαθβ is given by the integral A=12βα[f(θ)]2dθA=12βα[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 αθβαθβ is given by the integral L=βα[f(θ)]2+[f(θ)]2dθ=βαr2+(drdθ)2dθ.

Key Equations

  • Area of a region bounded by a polar curve

    A=12βα[f(θ)]2dθ=12βαr2dθ
  • Arc length of a polar curve

    L=βα[f(θ)]2+[f(θ)]2dθ=βαr2+(drdθ)2dθ