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Polar Area Between Curves
AP Calculus BC · Unit 9 Polar Coordinates
Learning Objectives
Set up ½∫(R² − r²) dθ for enclosed area
Find intersection points of polar curves
Identify which curve is outer (R) vs inner (r)
Correctly handle negative r values
Area Formulas
Single Curve Area:
A = ½ ∫ [r(θ)]² dθ
Area Between Two Curves:
A = ½ ∫ [R(θ)² − r(θ)²] dθ
where R(θ) is the outer curve
and r(θ) is the inner curve.
Tags
polar area
between curves
integration
CURVE PAIR
Circle r=3 vs Cardioid r=1+2sinθ
Rose r=sin(2θ) vs Circle r=1
Limaçon r=2+cosθ vs Circle r=3
Two Roses: r=cos(2θ) vs r=sin(2θ)
Integration From α
0.00π
Integration To β
1.00π
Area (Outer)
—
Area (Inner)
—
Area Between = ½∫(R²−r²)dθ
—
Cyan
: Curve 1 (R, outer)
Red
: Curve 2 (r, inner)
Yellow shading
: Area between curves from α to β