Area in Polar Coordinates

Calculus 2 · Axiom Academy

LESSON Area in Polar Coordinates Why polar area integrates ½r², not r — it is the geometry of a thin sweeping wedge. Pick a polar curve — here the cardioid — and isolate a thin slice between angle and . Its two straight edges are both radii of length , and over a tiny angle the curved outer edge is nearly a circular arc. So the slice is almost a circular sector : radius r , angle . A full circle is the sector that sweeps all of A sector keeps only the fraction of it 2. Sweep the Wedge: Area Accumulates To get the whole region, let the wedge sweep from to . At each angle it lays down its little , and the running sum of those sectors is the area swept so far. Watch the wedge orbit the cardioid while the live total climbs — for the full 0 to it settles at exactly . where traces the curve from to Area — a single sector at angle . Adding sectors as turns the sum into the integral. The slice is a triangle-like sector, so its area scales with r^2 . 3. Watch the Limits: Sweeping Twice The formula is only as honest as its limits. The curve is a circle of radius — but as runs from 0 to the wedge sweeps that whole circle twice : once while r > 0 , then again over the same region while r < 0 . Integrate blindly over and the running total reaches — exactly double the circle's true area . The circle is traced twice, so its area is counted twice. One full tracing — the correct area . Sketch the curve first and find where r returns to 0 . Integrate over one complete tracing of the region.

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