Arc length of a curve in polar coordinates
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Arc length of a curve - Parametric form
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The expression y = ax2 describes a parabola as a relationship between the Cartesian coordinates x and y. The expression [Equation goes here - download the original to see it.] describes a cardioid as a relationship between polar coordinates r and q. It is often more convenient to describe a curve in parametric form. In parametric form we specify the coordinates of a point on the curve by separate functions of another variable, called a parameter. [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] is the position function. When the parameter takes the value t, the position is: [Equation goes here - download the original to see it.] For example, a circle of radius 2 is given in parametric form by [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.] We now need to derive the formula for the arc length when the equation is given in parametric form: [Diagram goes here - download the original to see it.] We approximate the length of the curve by line segments. As the parameter increases by [Equation goes here - download the original to see it.] the x-coordinate changes from [Equation goes here - download the original to see it.] and the y-coordinate from [Equation goes here - download the original to see it.] Then the line segment is: [Equation goes here - download the original to see it.] In the limit as [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] But the curve is approximated by the sum of the line segment. Hence, the arc length, s, is given by: [Equation goes here - download the original to see it.] Example Find the length of the circumference of a circle of radius a using the parametric form. Solution [Equation goes here - download the original to see it.]
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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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