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 = img, where img or img
  • Arc length (L) of a curve in parametric form between t = a and t = b is given as L = img, where img
  • Arc length (L) of a curve in polar form between θ = a and θ = b is given as L = img, where img

Examples of Arc Length

  • The arc length(L) of the parametric curve for x = 1 - sin t, y = 2 + cos t, img ≤ t ≤ imgis L = img [img = - cos t, img = - sin t]
    = img
    = img, since cos2 t + sin2 t = 1
    = img , by integrating
    = img = Π units.

Video Examples: Arc Length


Solved Example Clockple on Approximate

Ques: Find the arc length of the polar curve r = e, 0 ≤ θ ≤ 1.

    Choices:
    A. img(1 - e2) units
    B. eimg units
    C. img(e2- 1) units
    D. (e2 - 1) units
    Correct Answer: C

Solution:

    Step 1: r = e, 0 ≤ θ ≤ 1 [Equation of the polar curve.]
    Step 2: img = 2e [Differentiate r with respect to θ.]
    Step 3: Arc length, Limg = [Use L =img .]
    Step 4: = img
    Step 5: = img
    Step 6: = img
    Step 7: = img img [Integrate.]
    Step 8: = img [e2 - eθ] [Substitute the limits.]
    Step 9: = img (e2- 1) [Simplify.]
    Step 10: Arc length of the given polar curve is img (e2- 1) units.
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