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The Area Below a Polar Curve

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Polar coordinates provide an alternate way of specifying a point in the plane. The polar coordinates [r, θ] represent the point at a distance r from the origin, rotated θ radians counterclockwise from the positive x -axis. Since r represents a distance, it is typically positive. Sometimes, r is allowed to be negative; in this case [r, θ] represents the reflection about the origin of the point [| r|, θ] .

It follows from basic trigonometry that the point with polar coordinates [r, θ] has Cartesian coordinates

(r cosθ, r sinθ)    

Figure %: Point with Polar Coordinates [r, θ]

Going the other direction, the point with Cartesian coordinates (x, y) has polar coordinates

, tan-1    

if it lies in quadrants I or IV and polar coordinates

, tan-1 + Π    

if it lies in quadrants II or III .

A polar function r(θ) has a graph consisting of the points [r(θ), θ] . Such a graph is known as a polar curve. One of the simplest polar curves is the circle, the graph of the polar function r(θ) = c , for some constant c . In the remainder of this section, we investigate how to find the area enclosed by a polar curve from one value of θ to another. For example, we might wish to find the area of the region below the curve r(θ) = 1 from θ = 0 to θ = Π/2 (this region is of course a quarter of the interior of a unit circle).