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example for integration in polar coordinates

In this section, we define polar coordinates and study their relation to Cartesian coordinates. One of the distinctions between polar and Cartesian coordinates is that while a point in the plane has just one pair of Cartesian coordinates, it has infinitely many pairs of polar coordinates. This has interesting consequences for graphing, as we shall see in the next lectures. Polar coordinates enable us to describe all conic sections with a single equation, and the calculus we have done in rectangular coordinates carries over to this new system as well.
To define polar coordinates, we first fix an origin 0 and an initial ray from 0. then each point p can be located by assigning to it a polar coordinate pair, in which the first number, r , gives the directed distance from 0 to P and the second number, is the angle from the initial ray to the segment 0P.

As in trigonometry , the angle is positive when measured counter- clockwise  and negative when measured clockwise . but the angle associated with a given point is not unique

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