Polar Calculus

Part 1 of 7 — Polar Coordinates & Slopes

Polar ↔ Rectangular Conversion

$x = rcos heta, quad y = rsin heta$ $r^2 = x^2 + y^2, quad an heta = y/x$

Slope of a Polar Curve

For $r = f( heta)$:

rac{dy}{dx} = rac{rac{dr}{d heta}sin heta + rcos heta}{rac{dr}{d heta}cos heta - rsin heta}

(This comes from the chain rule: $x = rcos heta$, $y = rsin heta$)

Polar Slopes 🎯

Key Takeaways — Part 1

The slope formula uses the product rule on $x = r\cos\theta$ and $y = r\sin\theta$.

Polar Calculus

Part 2 of 7 — Area in Polar Coordinates

Area Formula

A = rac{1}{2}int_alpha^eta r^2,d heta

Worked Example: Cardioid

$r = 1 + cos heta$, find total area.

A = rac{1}{2}int_0^{2pi}(1 + cos heta)^2,d heta

Expand: (1 + cos heta)^2 = 1 + 2cos heta + cos^2 heta = 1 + 2cos heta + rac{1+cos 2 heta}{2}

= rac{3}{2} + 2cos heta + rac{cos 2 heta}{2}

Polar Area 🎯

Key Takeaways — Part 2

1. Area = $\frac{1}{2}\int r^2\,d\theta$
2. Choose bounds carefully — trace the curve once!

Polar Calculus

Part 3 of 7 — Area Between Polar Curves

Area Between Two Polar Curves

Area Between Curves 🎯

Key Takeaways — Part 3

For area between curves: $\frac{1}{2}\int(r_{\text{outer}}^2 - r_{\text{inner}}^2)\,d\theta$.

Polar Calculus

Part 4 of 7 — Common Polar Curves

Gallery of Polar Curves

Polar Curve ID 🎯

Key Takeaways — Part 4

Know the standard polar curves for the AP exam.

Polar Calculus

Part 5 of 7 — Tangent Lines at the Pole

When Does the Curve Pass Through the Origin?

Set $r = 0$ and solve for $heta$. Each solution gives a direction through the origin, and that value of $heta$ gives the angle of a tangent line at the origin!

Example

$r = sin(2 heta)$: $r = 0$ when $2 heta = npi$, so $heta = 0, pi/2, pi, 3pi/2$.

The tangent lines at the origin are $heta = 0$ ($x$-axis) and $heta = pi/2$ ($y$-axis).

Tangent Lines at Pole 🎯

Key Takeaways — Part 5

At the origin: $r = 0$ gives the tangent lines.

Polar Calculus

Part 6 of 7 — Practice Workshop

Mixed Polar Practice 🎯

Workshop Complete!

Polar Calculus — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Polar Calculus — Complete! ✅