ARC LENGTH

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

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 = 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 - e^θ] [Substitute the limits.]
Step 9: =  (e²- 1) [Simplify.]
Step 10: Arc length of the given polar curve is  (e²- 1) units.

Quick Summary

Arc length is the length of a curve.
Formulas vary depending on how the curve is represented (rectangular, parametric, polar).
Integration is used to calculate arc length.

\[ L = \int_{a}^{b} \sqrt{1 + (\frac{dy}{dx})^2} dx \]

🍎 Teacher Insights

Emphasize the geometric interpretation of arc length as the limit of a sum of small line segments. Use examples from different areas of mathematics and physics to illustrate the applications of arc length.

🎓 Prerequisites

Integration
Differentiation
Parametric Equations
Polar Coordinates

Check Your Knowledge

Q1: The arc length of the parametric curve for x = 1 - sin t, y = 2 + cos t, 0 ≤ t ≤ π/2 is:

π/2 π 2π π/4

Q2: Find the arc length of the polar curve r = e^(2θ), 0 ≤ θ ≤ 1.

(sqrt(5)/2)(1 - e^2) units e*sqrt(5) units (sqrt(5)/2)(e^2 - 1) units (e^2 - 1) units

Frequently Asked Questions

Q: How do I choose the correct formula for arc length?
A: Identify whether the curve is given in rectangular, parametric, or polar form, and then use the corresponding formula.

Q: What are the units of arc length?
A: The units of arc length are the same as the units of the coordinate system (e.g., meters, feet, units).