Polar curves and areas

What you'll learn

1. 1

   Imagine drawing a curve by spinning a rope around a point — that's a polar curve! Each point is described by how far away it is (r) and the angle you've spun (θ).

2. 2

   In polar coordinates, what do the two numbers (r, θ) tell you?

3. 3

   Here's a polar curve: r = 2 + 2 cos(θ). It's a cardioid — shaped like a heart! The distance r changes as you go around the circle.

4. 4

   Let's find the area inside one loop of r = 2 sin(2θ) — a four-leaf clover!

5. 5

   Drag the slider to see how the area of a sector changes as θ increases. The sector is the region between two angles, like a pizza slice.

6. 6

   What is the formula for the area enclosed by a polar curve from θ = a to θ = b?

7. 7

   For r = 2 sin(2θ), one loop goes from θ = 0 to θ = π/2. What is the total area of ALL 4 loops?