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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 = ( x2 - y2 )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: ℝ2 → ℝ2; 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 = ʃʃF(R) 1 dA = ʃʃR r dr dθ = ʃ02πʃr2π r dθ dr = …