Curve

R

$\mathbf{R}\prime$

Arc-Length Function

$y\=f\left(x\right)$

$\left[\begin{array}{c}x\\ f\(x\)\end{array}\right]$

$\left[\begin{array}{c}1\\ f\prime \(x\)\end{array}\right]$

$s\left(x\right)\={\int }_{x_0}^x\sqrt{1\+{\left(f\prime \left(u\right)\right)}^2}\mathit{\ⅆ}u$

$\left\{\begin{array}{c}x\=x\left(p\right)\\ y\=y\left(p\right)\end{array}\right\}$

$\left[\begin{array}{c}x\(p\)\\ y\(p\)\end{array}\right]$

$\left[\begin{array}{c}x\prime \(p\)\\ y\prime \(p\)\end{array}\right]$

$s\left(p\right)\={\int }_{p_0}^p\sqrt{{\left(x\prime \left(u\right)\right)}^2\+{\left(y\prime \left(u\right)\right)}^2}\mathit{\ⅆ}u$

$\left\{\begin{array}{c}x\=x\left(p\right)\\ y\=y\left(p\right)\\ z\=z\left(p\right)\end{array}\right\}$

$\left[\begin{array}{c}x\(p\)\\ y\(p\)\\ z\(p\)\end{array}\right]$

$\left[\begin{array}{c}x\prime \(p\)\\ y\prime \(p\)\\ z\prime \(p\)\end{array}\right]$

$s\left(p\right)\={\int }_{p_0}^p\sqrt{{\left(x\prime \left(u\right)\right)}^2\+{\left(y\prime \left(u\right)\right)}^2\+{\left(z\prime \left(u\right)\right)}^2}\mathit{\ⅆ}u$

Table 2.2.1   The arc-length function

Example 2.2.1

Calculate the length of the helix defined in Example 2.1.4.

Example 2.2.2

Calculate the length of the curve defined in Example 2.1.5.

Example 2.2.3

Calculate the length of the curve defined parametrically by $x\left(t\right)\=t \cos \left(t\right)$, $y\left(t\right)\=t \sin \left(t\right)$, $z\left(t\right)\=t^3\/6$ for $0\le t\le 2 \mathrm{\π}$.

Example 2.2.4

Obtain $s\left(t\right)$, the arc-length function for the helix in Example 2.1.4.

Example 2.2.5

Obtain $s\left(t\right)$, the arc-length function for the curve in Example 2.1.3.

Example 2.2.6