Polar Area

The central idea

For a polar curve $r=f(\theta)$ traced once from $\alpha$ to $\beta$, the signed swept area is $A=\frac12\int_\alpha^\beta r^2d\theta$. Area between an outer and inner curve is $\frac12\int(R^2-r^2)d\theta$ where their radial ordering is valid. Bounds must avoid unintended repeated tracing.

Definitions, hypotheses, and notation

Intersection equations may include both equal radii and representations shifted by $\pi$ because polar coordinates are nonunique. For many textbook curves, a careful sketch and symmetry identify the desired loops more reliably than solving $r_1=r_2$ alone. A zero of $r$ marks passage through the pole and often separates loops.

The formula computes geometric area when bounds trace the intended radial region once. Orientation reversal changes the sign of $d\theta$ and of the integral, so conventional geometric area uses increasing bounds or an absolute correction after interpreting the tracing.

Conceptual meaning

A narrow polar sector with radius $r$ and angle $d\theta$ has area approximately $r^2d\theta/2$. Squaring radius makes negative $r$ contribute positive sector area, but negative radius can alter where the point lies and how the curve is traced.

A dependable method and decision rule

1. Sketch the curve and identify symmetry and tracing interval.
2. Solve intersections or zeros that set angular bounds.
3. Determine outer and inner radius on each angular interval.
4. Integrate one half of the squared-radius difference.
5. Use symmetry only after verifying the chosen sector repeats congruently.

Fully worked example

Graphical or geometric meaning

Rays from the pole sweep through the region. At each angle, the outer radial segment determines a thin sector. For area between curves, removing the inner sector leaves a curved polar washer.

Common mistakes and why they fail

Verification and reasonableness checks

• Compare with a known Cartesian area when the curve is recognizable.
• Trace key angles and mark negative-radius behavior.
• Confirm symmetry multiplier and base interval cover the region exactly once.

A sweep contributes a sector

A small angle contributes approximately $\tfrac12r^2\,d\theta$, giving $A=\tfrac12\int_\alpha^\beta r(\theta)^2\,d\theta$. Determine $\alpha$ and $\beta$ from how the curve is traced, not from familiar Cartesian bounds. Because negative $r$ plots in the opposite direction, inspect the actual region and solve intersections carefully. Between curves, integrate $\tfrac12(R^2-r^2)$ only where $R$ is genuinely outer; split when the order changes or a curve retraces. Symmetry can shorten work, but multiply a smaller sector only after verifying that its rotated copies cover the region exactly once. A sketch with a few labeled angles is often the best defense against double counting.

Practice

1. Find the area inside $r=1$.
2. Find the area for $r=2$, $0\le\theta\le\pi/2$.
3. What is the sector integrand?

Connections and next steps

• Polar Coordinates in Calculus
• Area Between Curves
• Double Integrals in Polar Coordinates