Areas in polar coordinates
A polar curve requires both a radial equation and an angular interval.
The signed polar coordinate r may represent a point in the opposite direction when r is negative. Sketch or inspect the parameter interval before choosing area limits; tracing a loop twice can double the integral.
An area swept once between θ = α and θ = β is one half the integral of r² with respect to θ. Angles must be in radians. To find an area between curves, identify which boundary lies farther from the origin throughout each interval.
Worked example
Find the area of the quarter-circle r = 2 for 0 ≤ θ ≤ π/2.
- Use A = (1/2)∫r² dθ.
- Since r² = 4, integrate 2 from 0 to π/2.
Answer: π
Revise first: Integration and exact area.
Choose an interval that traces the region once
The polar point (r, θ) has Cartesian coordinates (r cos θ, r sin θ). A negative r puts the point on the opposite ray; it is not a negative distance. A curve can retrace a region as θ increases, so a full-turn integral is not automatically the area of the union.
For a region swept once between two radial bounds, area = ½∫r² dθ. Find where r is zero, inspect the trace direction and decide which bounds enclose the requested region before integrating. For a difference of polar regions, establish which curve is outer over each interval.
One petal of r = 4 sin(3θ)
- Consecutive zeros at θ = 0 and π/3 enclose one petal with nonnegative radius.
- Area = 8∫₀^(π/3) sin²(3θ) dθ.
- Use sin²(3θ) = (1 − cos(6θ))/2. The sine endpoint terms vanish.
- The area is 4π/3. Integrating over 0 to 2π traces every petal twice for this odd-frequency rose.
Does r = −2 at θ = 0 place the point at (2, 0)?
No. It is at (−2, 0), since x = r cos θ and y = r sin θ.
Take the idea further
Compare r = a sin(kθ) and r = a cos(kθ). The petals rotate, but the area of one petal remains a²π/(4k).
Practise areas in polar coordinates
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 9FM0 · core · 7 Polar coordinates: Areas in polar coordinates
- 7367 · core · G Polar coordinates: Areas in polar coordinates
- H245 · core · 7 Polar coordinates: Areas in polar coordinates
- H645 · core · Polar coordinates: Areas in polar coordinates
- 9FM0 · core · 7 Polar coordinates: Area swept by constant polar radius
- 7367 · core · G Polar coordinates: Area swept by constant polar radius
- H245 · core · 7 Polar coordinates: Area swept by constant polar radius
- H645 · core · Polar coordinates: Area swept by constant polar radius
- 9FM0 · core · 7 Polar coordinates: An area swept by a limaçon
- 7367 · core · G Polar coordinates: An area swept by a limaçon
- H245 · core · 7 Polar coordinates: An area swept by a limaçon
- H645 · core · Polar coordinates: An area swept by a limaçon
- 9FM0 · core · 7 Polar coordinates: Area of one polar rose petal
- 7367 · core · G Polar coordinates: Area of one polar rose petal
- H245 · core · 7 Polar coordinates: Area of one polar rose petal
- H645 · core · Polar coordinates: Area of one polar rose petal
- 9FM0 · core · 7 Polar coordinates: Area swept by a polar spiral
- 7367 · core · G Polar coordinates: Area swept by a polar spiral
- H245 · core · 7 Polar coordinates: Area swept by a polar spiral
- H645 · core · Polar coordinates: Area swept by a polar spiral
Next practice: Sketching functions.