Polar Area Lab — Fibo Math Lab

Fibo Math Lab

Polar Area Lab

Skill goal: Compute areas with the polar area formula A = ½∫r² dθ.

1 · Observe

The full curve is drawn in gray. As you sweep θ from 0 to 2π, teal sectors fill in — each one a thin slice of the total area. The gold spoke is the radius r(θ) at the current angle.

Drag around the center (or use the slider) to sweep θ from 0 to 2π.

2 · Manipulate

Area accumulates live via a fine Riemann sum (720 steps) of ½r² Δθ. Sweep all the way to 2π and compare with the analytic total.

3 · Predict

Before sweeping: what is the total area of the cardioid r = 1 + cos θ? Make a guess, then sweep to 2π and reveal.
Total cardioid area = 3π/2 ≈ 4.712. Sweep to θ = 2π above and the numeric sum lands right on it.

4 · See

θ swept
0.50π
Swept area (numeric)
—
Analytic total
4.712

Keep sweeping — the numeric sum should approach the analytic total.

5 · Explain

Why it works: a thin sector of angle Δθ is almost a triangle with area ≈ ½·r·r·Δθ = ½r²Δθ. Adding up all the sectors from 0 to 2π gives A = ½∫r² dθ.

Cardioid check: A = ½∫₀2π (1 + cos θ)² dθ
(1 + cos θ)² = 1 + 2cos θ + cos²θ = 3/2 + 2cos θ + (cos 2θ)/2
∫₀2π = 3π + 0 + 0 = 3π  →  half of that is 3π/2 ≈ 4.712 ✓

6 · Challenge

1. A circle with r = 3. Use A = ½∫₀2π r² dθ. Enter the exact area (within 0.01).

2. The circle r = 2 sin θ is fully traced as θ goes from 0 to π. Enter its area (within 0.01).

3. The rose r = sin 3θ: evaluate ½∫₀2π sin²(3θ) dθ. Enter the area (within 0.01).

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