Arc Length
Length of a curve.
Key idea
Length of a curve.
In depth
L = ∫ₐᵇ √(1 + (f′(x))²) dx comes from adding up tiny hypotenuses √(dx² + dy²). Most integrals are hard to evaluate exactly, so numeric methods are common.
Worked example
y = x^(3/2) on [0, 1]: L = ∫√(1 + (9/4)x) dx = (8/27)((13/4)^(3/2) − 1) ≈ 1.44.
Formula
Real-world uses
- Measuring the length of a cable hanging between two poles or a curved road
Common mistake
Forgetting the 1 + under the root.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Arc Length?
Length of a curve. It is a topic at the expert level in Calculus.
What is the formula for Arc Length?
The key formula is L = ∫ √(1+f′²) dx.
Where is Arc Length used in real life?
Real-world examples: Measuring the length of a cable hanging between two poles or a curved road.
What should I learn before Arc Length?
Make sure you are comfortable with Volumes by Shells, the previous topic in Calculus.
What is a common mistake with Arc Length?
Forgetting the 1 + under the root.
How can I practise Arc Length?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Arc Length?
Move on to Surface Area of Revolution next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.