Polar Calculus

Key idea

Area and slope in polar coordinates.

In depth

In polar coordinates, area is A = ½∫ r² dθ and arc length is ∫√(r² + (dr/dθ)²) dθ. Sectors, not rectangles, are the building blocks.

Worked example

Area of r = 2 cos θ (a circle of radius 1): ½∫ from −π/2 to π/2 of 4cos²θ dθ = π.

Formula

Real-world uses

• Finding areas inside petal-shaped curves and the slopes of spiral paths in engineering

Common mistake

Using ∫r dθ for area, or the wrong limits for roses and loops.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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