Arc Length
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Definition of Arc Length
Arc Length is the length of an arc or curve.
More About Arc Length
 Arc length (L) of a curve in rectangular form between x = a and x = b is given as L = , where or
 Arc length (L) of a curve in parametric form between t = a and t = b is given as L = , where
 Arc length (L) of a curve in polar form between θ = a and θ = b is given as L = , where
Examples of Arc Length
 The arc length(L) of the parametric curve for x = 1  sin t, y = 2 + cos t, ≤ t ≤ is L = [ =  cos t, =  sin t]
=
= , since cos^{2} t + sin^{2} t = 1
= , by integrating
= = Π units.
Video Examples: Arc Length
Solved Example Clockple on Approximate
Ques: Find the arc length of the polar curve r = e^{2θ}, 0 ≤ θ ≤ 1.
Choices:
A. (1  e^{2}) unitsB. e units
C. (e^{2} 1) units
D. (e^{2}  1) units
Correct Answer: C
Solution:

Step 1: r = e^{2θ}, 0 ≤ θ ≤ 1 [Equation of the polar curve.]
Step 2: = 2e^{2θ} [Differentiate r with respect to θ.]
Step 3: Arc length, L = [Use L = .]
Step 4: =
Step 5: =
Step 6: =
Step 7: = [Integrate.]
Step 8: = [e^{2}  e^{θ}] [Substitute the limits.]
Step 9: = (e^{2} 1) [Simplify.]
Step 10: Arc length of the given polar curve is (e^{2} 1) units.
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