Length of a curve

By Li Du (P4526) on Friday, May 25, 2001 - 01:22 pm :

Hi all,

I was trying to derive the formula for the surface area of a sphere using integration. And I had this problem:
How to calculate the length of a curve given the boundaries? (eg. the length of y=ln(x) for x between 3 and 10 etc) A calculus approach is appreciated.

So in fact I want to ask two questions: the surface area of a sphere and the length of a curve.

Thanks in advance!

Ryan

By Geoff Milward (Gcm24) on Friday, May 25, 2001 - 02:08 pm :

To find the length of a curve one forms the integral

$\int dx = \int ({dx}^{2} + {dy}^{2})^{1 / 2} = \int (1 + (dy / dx)^{2})^{1 / 2} dx$

over the appropriate limits.

So $y = \ln x$ would be

$\int (1 + 1 / x^{2})^{1 / 2}$

and typically these integrals are rather hard.

The surface area of a sphere is just the integral of the spherical volume element over $θ$ and $ϕ$ , most standard calculus books will have this.

Hope this helps

Geoff

By Barkley Bellinger (Bb246) on Friday, May 25, 2001 - 02:09 pm :

If you consider the length of a very small piece of curve in the

x

y

plane to be

ds

, the Pythagoras' theorem

\Rightarrow {ds}^{2} = {dx}^{2} + {dy}^{2} = {dx}^{2} (1 + (dy / dx)^{2})

. So since the length of the whole curve is

\int ds

(with appropriate limits), this is just

\int (1 + (dy / dx)^{2})^{1 / 2} dx

, with the corresponding limits.