∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 6 Theory

Parametric Curves & Polar Coordinates

Rose petals, cardioids, arc length, and swept angular area

Trace polar curves r = f(θ) like multi-petaled roses and calculate enclosed sector areas using (1/2) ∫ r(θ)² dθ.

1. Polar Sector Area vs. Cartesian Area

In Cartesian coordinates, area elements are rectangular dA = dx dy. In polar coordinates, rotating an increment dθ sweeps out a circular wedge of radius r and arc length r dθ, giving area dA = (1/2) r² dθ.

dA=12r2 dθdA = \frac{1}{2} r^2\,d\theta

Quick Reference & Cheat Sheet

Polar Arc Length Formula
L=∫αβr2+(drdθ)2 dθL = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta
Pythagorean theorem on radial dr and angular r dθ displacements.