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 cos2 t + sin2 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 = e2θ, 0 ≤ θ ≤ 1.
Choices:A. (1 - e2) units
B. e units
C. (e2- 1) units
D. (e2 - 1) units
Correct Answer: C
Step 1: r = e2θ, 0 ≤ θ ≤ 1 [Equation of the polar curve.]
Step 2: = 2e2θ [Differentiate r with respect to θ.]
Step 3: Arc length, L = [Use L = .]
Step 4: =
Step 5: =
Step 6: =
Step 7: = [Integrate.]
Step 8: = [e2 - eθ] [Substitute the limits.]
Step 9: = (e2- 1) [Simplify.]
Step 10: Arc length of the given polar curve is (e2- 1) units.