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Chapter 9: Problem 61

Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.

Short Answer

Expert verified
The arc length formula for a polar curve is: \[ L = \int \sqrt{ [r'(胃)]^{2} + [r(胃)]^{2} } d胃 \], where \( r(胃) \) is the equation in polar coordinates.

Step by step solution

01

Statement of Arc Length for a Curve in Parametric Form

The arc length formula for a curve defined parametrically by \(x=f(t)\) and \(y=g(t)\) for \(a\leq t \leq b\) is given by: \[ L = \int_{a}^{b} \sqrt{ [f'(t)]^{2} + [g'(t)]^{2} } dt \] where \(f'(t)\) and \(g'(t)\) are the derivatives of \(f(t)\) and \(g(t)\) respectively.
02

Conversion of Polar Coordinates to Parametric Form

The relationship between polar and Cartesian coordinates offers a connection between a polar curve and parametric equations. Specifically, if \( r = f(胃)\) is a polar curve, this can be converted into the parametric equations \(x(t) = f(t)\cos(t)\), \(y(t) = f(t)\sin(t)\). The range for \(t\) will be the same range of \(胃\) described in the polar coordinate.
03

Derivation of the Arc Length for a Polar Curve

Substituting the parametric equations into the arc length formula we obtain: \[ L = \int \sqrt{ [f(t)\cos(t)']^{2} + [f(t)\sin(t)']^{2} } dt \]. Simplifying, we get: \[ L = \int \sqrt{ [f'(t)cos(t) - f(t)sin(t)]^{2} + [f'(t)sin(t) + f(t)cos(t)]^{2} } dt \]. Finally, we arrive at the formula for the length of a polar curve: \[ L = \int \sqrt{ [r'(胃)]^{2} + [r(胃)]^{2} } d胃 \], where \( r(胃) \) is the equation in polar coordinates.

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