🌸
Polar Curves & Area
AP Calculus BC · Unit 9 Parametric/Vector/Polar
Learning Objectives
Polar Coordinates:
Position is defined not by $(x,y)$ but by angle $\theta$ and radius $r(\theta)$ from the origin.
Area Integrals:
Cartesian area uses rectangles ($\int y dx$). Polar area uses circular wedges: $A = \frac{1}{2}\int r^2 d\theta$.
Negative Radii:
If $r(\theta) < 0$, the point is plotted exactly $180^\circ$ backward through the origin! This is how inner loops of limaçons form.
Tags
Polar
Roses
Cardioids
Area
POLAR FUNCTION
$r(\theta)$ =
1 + \cos(\theta)
Select Polar Curve
Cardioid (Heart)
$1 + \cos(\theta)$
Rose Curve (4-Petal)
$\sin(2\theta)$
Inner-loop Limaçon
$1 + 2\sin(\theta)$
Sweep Angle ($\theta$)
$\theta$ Limit =
3.14
$\pi$
Polar Area Integration
$Area = \frac{1}{2} \int_{0}^{\theta} [r(\theta)]^2 d\theta$
INSTANTANEOUS VALUES
Current Radius $r =$
0.00
Wedge $dA =$
0.00
TOTAL SWEPT AREA
Area $A =$
0.00
Cardioid:
At $\theta=0$, $r=2$. As $\theta$ moves towards $\pi$, $\cos$ becomes $-1$, making the radius collapse beautifully to $0$ at the origin, creating the heart cleft shape.