Calculus with Polar Coordinates

Ex. Find all points of intersection of r = 1 – 2cos θ and r = 1.

The area bounded by r = f (θ), α ≤ θ ≤ β is

212A f d

Ex. Find the area of one petal of r = 3cos 3θ

6

4

2

- 2

- 4

- 6

- 5 5

Ex. Find the area between the inner and outer loops of the curve r = 1 – 2sin 3θ

6

4

2

- 2

- 4

- 6

- 5 5

Ex. Find the area inside r = 3sin θ and outside r = 1 + sin θ.

6

4

2

- 2

- 4

- 6

- 5 5

The arc length of the curve r = f (θ), α ≤ θ ≤ β is

2 2s f f d

Ex. Find the length of the curve given by r = 1 + sin θ

Pract.

1. Find the area of one loop of r = cos 2θ.

2. Find the area of the region that lies inside r = 3sin θ and outside r = 1 + sin θ.

3. Set up an integral to find the length of the curve r = θ for 0 ≤ θ ≤ 2π.

8

22

0

1 d