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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).

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