9.8 Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Area of a Polar Region Formula

For a polar curve r = f(θ), the area of the region swept out from θ = a to θ = b is A = (1/2)∫[a,b] r² dθ. This comes from summing infinitely many thin circular sectors, each with area (1/2)r²dθ. Always identify the correct bounds where the curve traces the region exactly once.

Finding the Right Bounds

To find θ-bounds, determine where the curve starts/ends or where r = 0 (the curve passes through the origin). For a full closed curve like a circle or rose petal, find the interval where r stays one sign and the shape traces once without overlap. Graphing helps confirm the correct interval before integrating.

Area Between Two Polar Curves

If one curve is inside another over [a,b], the area between them is A = (1/2)∫[a,b] (R² − r²) dθ, where R is the outer and r is the inner radius. Find intersection points by setting the two r-expressions equal to determine integration limits, and check for symmetry to simplify computation.

Key terms

Area of a polar region formula

A = (1/2)∫[a,b] r² dθ, where r = f(θ).

How to find bounds for one petal/loop

Solve r = 0 to find where the curve passes through the origin, marking the start and end of the loop.

Area between two polar curves

A = (1/2)∫[a,b] (R² − r²) dθ, where R is the outer curve and r is the inner curve.

Why polar area uses sectors, not rectangles

Polar coordinates naturally divide regions into thin circular sectors, each with area (1/2)r²dθ, rather than rectangular strips.

Key check before integrating polar area

Graph the curve to confirm the θ-interval traces the desired region exactly once, avoiding overlap or missing area.

Try an AP-style question

Which formula gives the area of a polar region?

1. A = (1/2)∫ r² dθ
2. A = ∫ r dθ
3. A = (1/2)∫ r dθ
4. A = π∫ r² dθ