# Length of a curve

#### soandos

is there a way to find the length of a curve between two x values?
if so, what is it.
thanks

#### jeffreydk

Yes since the element of length is (assuming two dimensions)

$$ds=\sqrt{dx^2+dy^2}$$

If you integrate you get the length of the curve $s$ from $[a,b]$

$$s=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx$$

#### HallsofIvy

Homework Helper
In three dimensions, write x, y, z as functions of parameter t and do the same:
$$\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2+ \left(\frac{dz}{dt}\right)^2}dt$$

#### GoodMax

I am sorry for my broken english.

How can one find the length of a curve, if the coordinate system is not rectangular (for instance, it is spherical)?

Please if not inconvenient to you, point out my mistakes in my english.

#### HallsofIvy

For example, suppose a path is given in spherical coordinates by $\rho= 1$, $\phi= \pi/3$, $\theta= t$, with parameter t. In Cartesian coordinates that is $x= \rho cos(\theta) sin(\phi)= (\sqrt{3}/2)cos(t)$, $y= \rho sin(\theta)sin(\phi)= (\sqrt{3}/2)sin(t)$, $z= \rho cos(t)= cos(t)$.