key term - Arc length

Definition

Arc length is the distance measured along the curve between two points. It is calculated by integrating the square root of the sum of the squares of derivatives of the function defining the curve.

5 Must Know Facts For Your Next Test

1. The formula for arc length in Cartesian coordinates is $$L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$.
2. For parametric equations, the arc length formula is $$L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$.
3. In polar coordinates, the arc length can be found using $$L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$$.
4. Arc length calculation often involves using substitution and trigonometric identities to simplify integrals.
5. Understanding how to derive and use these formulas is crucial for problems involving curves and surface areas.

Review Questions

Related terms

Integral:

A mathematical concept that represents the area under a curve; essential for calculating arc lengths.

Parametric Equations:

Equations where the coordinates are expressed as functions of a parameter, useful in determining arc lengths and surface areas.

Surface Area:

$$A = \int_a^b (2\pi y) ds$$, where ds represents an infinitesimal element of arc length; related to finding surface area by revolution.