# Length of a curve

1. Dec 22, 2008

### soandos

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

2. Dec 22, 2008

### 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 $[a,b]$

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

3. Dec 22, 2008

### HallsofIvy

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

4. Dec 23, 2008

### 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.

5. Dec 23, 2008

### HallsofIvy

Staff Emeritus
By using exactly the same formulas but converting from the other coordinate system.

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 $x= \rho cos(\theta) sin(\phi)= (\sqrt{3}/2)cos(t)$, $y= \rho sin(\theta)sin(\phi)= (\sqrt{3}/2)sin(t)$, $z= \rho cos(t)= cos(t)$.

And my only criticism is that you should stop apologizing for your English. It is excellent. Far better than my (put whatever language you wish in here!).