Arc Length
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

Example 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.

Solved Example on Arc Length

Find the Arc Length of the polar Curve r = e^2θ, 0 ≤ θ ≤ 1.
Choices:
A. (1 - e²) units
B. e units
C. (e²- 1) units
D. (e² - 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: = [e² - e⁰] [Substitute the limits.]
Step 9: = (e²- 1) [Simplify.]
Step 10: Arc Length of the given polar Curve is (e²- 1) units.

Related Terms for Arc Length

• Arc
• Circle
• Circumference
• Curve
• Length
• Line
• Parametric Equations
• polar form
• Section

Additional Links for Arc Length

• Click here for samples
• Back to Mathematics Dictionary