Polar Coordinates Polar Coordinates: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ) Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 28 / 46
Polar Coordinates One can easily convert Cartesian coordinate (x, y) to polar coordinates (r, θ), and the other way around. r is a function of x, y; θ is also a function of x, y. r = x 2 + y 2 ; θ = tan 1 y x. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 29 / 46
Polar Coordinates x is a function of r and θ; y is a function of r and θ. x = r cos θ; y = r sin θ. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 30 / 46
Polar Coordinates Example 1. Convert the point with polar coordinates (2, π) to Cartesian coordinates. Solution: Since r = 2, θ = π, in Cartesian coordiates x = r cos θ = 2 cos π = 2; y = r sin θ = 2 sin π = 0. Thus the point in Cartesian coordinates is ( 2, 0). Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 31 / 46
Polar Coordinates Example 2. Convert the point with Cartesian coordinates ( 2, 2 3) to polar coordinates. Solution: Since x = 2, y = 2 3, in Cartesian coordiates r = x 2 + y 2 = 4 + 4 3 = 16 = 4; θ = tan 1 y x = tan 1 ( 3) = 4π/3. Thus the point in Cartesian coordinates is (4, 4π/3). Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 32 / 46
Polar Coordinates Polar curve: the graph of a polar equation r = f (θ), or F (r, θ) consist of all points P that satisfies the equation. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 33 / 46
Polar Coordinates Example 3. Convert the polar equation r = 3 to a Cartesian equation, and sketch the curve. Solution: Since r = x 2 + y 2, the Cartesian equation is x 2 + y 2 = 9. This is a circle of radius 3. Easy to sketch. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 34 / 46
Polar Coordinates Example 4. Convert the polar equation r = 3 sin θ to a Cartesian equation, and sketch the curve. Solution: r = 3 sin θ and y = r sin θ imply that r = 3 y r. Thus r 2 = 3y. To eliminate r, we write r = x 2 + y 2. Hence x 2 + y 2 = 3y. This is a Cartesian equation. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 35 / 46
Polar Coordinates What is this curve? Complete the square x 2 + (y 3 2 )2 = 9 4. Thus the curve is a circle, centered at (0, 3 2 ) with radius 3 2. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 36 / 46
Area formula revisited: Area between the graph of f (x) and x-axis is given by A = b a f (x) dx. olar coordinates 37 / 46
Area formula of the region between the graph of f (x) and g(x) is given by A = b a f (x) g(x) dx. olar coordinates 38 / 46
Let r = f (θ) for θ [a, b] be a curve in the plane. The polar region is the region enclosed by the ray θ = a, θ = b and the curve r = f (θ) for θ [a, b]. How to compute the area? A = b a 1 b 2 r 2 1 dθ = a 2 f 2 (θ)dθ. olar coordinates 39 / 46
Example 1. Given a polar curve r = 3 sin θ for θ [ π 4, 3π 4 ]. Compute the area of the polar region. Solution: By Example 4 in Chapter 10.3, polar curve r = 3 sin θ is a circle. 3π 4 A = π 4 = 9 4 3π 4 π 4 1 2 (3 sin θ)2 dθ 1 cos 2θdθ = 9 4 (π 2 1 2 sin 2θ 3π 4 π 4 ) (7) = 9 4 (π 2 + 1). polar coordinates 40 / 46
Area between two polar curves r = f (θ) and r = g(θ) for θ [a, b] is A = b a 1 2 f 2 (θ) 1 2 g 2 (θ)dθ. Example 2. Given a polar curve r = 2 sin θ and r = 1 + sin θ for θ [ π 4, 3π 4 ]. Compute the area of the polar region. polar coordinates 41 / 46
Solution: 3π 4 A = π 4 =. 1 2 (2 sin θ)2 1 2 (1 + sin θ)2 dθ (8) polar coordinates 42 / 46
Arc length in parametric curve L = b a (f (t)) 2 + (g (t)) 2 dt. polar coordinates 43 / 46
Polar curve r = f (θ) for θ [a, b] gives parametric equations: x = r cos θ = f (θ) cos θ; y = r sin θ = f (θ) sin θ with θ [a, b]. L = b a (x (θ)) 2 + (y (θ)) 2 dθ. polar coordinates 44 / 46
By chain rule, x (θ) = f cos θ f sin θ; Thus b L = = a b a y (θ) = f sin θ + f cos θ. (f cos θ f sin θ) 2 + (f sin θ + f cos θ) 2 dθ (f (θ)) 2 + f 2 (θ)dθ. (9) polar coordinates 45 / 46
Example 3. Find the length of the cardioid r = 1 + sin θ. Solution: 2π L = = = 0 2π 0 2π =... 0 (f (θ)) 2 + f 2 (θ)dθ (1 + sin θ) 2 + (cos θ) 2 dθ 2 + 2 sin θdθ (10) polar coordinates 46 / 46