Arc length

Definition

Arc length is the distance along a curve between two points. In calculus, it is calculated using integration techniques.

5 Must Know Facts For Your Next Test

1. The formula for arc length in Cartesian coordinates is $\int_a^b \sqrt{1 + (f'(x))^2} \, dx$.
2. For parametric equations, the arc length formula is $\int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$.
3. When dealing with polar coordinates, the arc length can be computed as $\int_a^b \sqrt{r(\theta)^2 + (r'(\theta))^2} \, d\theta$.
4. The arc length formula requires the derivative of the function representing the curve.
5. Arc length calculations often involve definite integrals.

Review Questions

• What is the general formula for finding the arc length of a function in Cartesian coordinates?
• How do you find the arc length of a curve given in parametric form?
• Explain how to compute the arc length of a curve when given in polar coordinates.

Related terms

Definite Integral: A type of integral that computes the net area under a curve within specified limits.

Parametric Equations: Equations where variables are expressed as functions of one or more independent parameters.

Surface Area: The total area that covers the surface of a three-dimensional object, which can also be calculated using integration techniques.