Example 1 Find the arc length of the circle r = 4cos θ

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Arc Length of Polar Curves and Area of Polar Regions MATHEMATICS 22 Institute of Mathematics University of the Philippines-Diliman 1 / 22
Arc Length of Polar Curves Recall: If C : x = x ( t ), y = y ( t ) where t [ a , b ], is a smooth parametric curve and is traced exactly once on [ a , b ], then the length of C is given by L = Z b a s dx dt 2 + dy dt 2 dt . Let C be a smooth polar curve with equation r = f ( θ ) traced exactly once when θ [ α , β ]. Then C is parametrized by x = f ( θ )cos θ and y = f ( θ )sin θ . Taking derivatives, we have dx d θ = f 0 ( θ )cos θ - f ( θ )sin θ = dr d θ · cos θ - r sin θ dy d θ = f 0 ( θ )sin θ + f ( θ )cos θ = dr d θ · sin θ + r cos θ 2 / 22
Arc Length of Polar Curves Taking the sum of the squares of the derivatives, we get dx d θ 2 + dy d θ 2 = r 2 + dr d θ 2 , which leads to the following formula. Arc Length Formula in Polar Coordinates If a smooth polar curve C : r = f ( θ ) is traced exactly once as θ varies from α to β , then the length of C is L = Z β α s r 2 + dr d θ 2 d θ . 3 / 22
Example 1 Find the arc length of the circle r = 4cos θ .
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Example 2 Find the length of the cardioid r = 1 + cos θ .
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