This site is being phased out.

# Polar coordinates

From Mathematics Is A Science

Jump to navigationJump to searchThese are the transition formulas for Cartesian coordinates and back:

( x, y ) ↦ ( r, θ ) x = r cos θ ↔ r = ( x^{2}- y^{2})^{1/2}y = r sin θ ↔ θ = arctan y / x.

## Integration

Find the area bounded by the curves:

r = θ r = 0 θ = 2π

Let's first plot these curves in in the ( r, θ )-plane.

Now, the "polar coordinate map" F transforms R into a region in the (x,y)-plane.

F: x = r cos(θ), y = r sin(θ); F: ℝ^{2}→ ℝ^{2}; F( r, θ ) = ( r cos(θ), r sin(θ) );

It is bounded by the curves:

1) line r = 0: F( 0, θ ) = ( 0, 0 ); 2) line θ = 2π: F( r, 2π ) = ( r cos(2π), r sin(2π) ) = ( r, 0 ), 0 ≤ r ≤ 2π; 3) line r = θ: F( θ, θ ) = ( θ cos(θ), θ sin(θ) ), 0 ≤ θ ≤ 2π.

θ is the angle with respect to the x-axis. Then

Area = ʃʃ_{F(R)}1 dA = ʃʃ_{R}r dr dθ = ʃ_{0}^{2π}ʃ_{r}^{2π}r dθ dr = …