Arc length of a curve in polar coordinates
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Arc length of a curve - in Polar coordinates
Some curves are more conveniently given in polar form - that is, as functions of the angle swept out from the x-axis (in an anti-clockwise direction) and the distance from the origin. [Diagram goes here - download the original to see it.] The function is expressed as a relationship between r and [Equation goes here - download the original to see it.]. Our task is to find the length of this curve between two points P and Q.[Diagram goes here - download the original to see it.]We derive this formula in the usual way by dividing the curve into segments and approximating each segment by a straight line.Consider one such argument of length [Equation goes here - download the original to see it.].[Diagram goes here - download the original to see it.]As the angle is increased by , the distance of the curve from the origin changes by from [Equation goes here - download the original to see it.]To find an expression for we construct a right-angled triangle, thus [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] Now PQ is approximately equal to the arc length given by [Equation goes here - download the original to see it.] This approximation would be exact if we took the limit; that is [Equation goes here - download the original to see it.] The length PZ is the change in the r coordinate, i.e. [Equation goes here - download the original to see it.] Hence, [Equation goes here - download the original to see it.] And in the limit the "infinitesimal" increase in arclength, ds, is given by: [Equation goes here - download the original to see it.] The arclength is given by the sum of small segments . Hence, [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] This is the arclength formula in polar coordinates.
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Contents of
Arc length of a curve in polar coordinates
1 Arc length of a curve - in Polar coordinates
2 Example- Arc length of a curve - in Polar coordinates
3 Arc length of a curve - Parametric form
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