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

òdx=ò(dx²+dy²)^1/2=ò(1+(dy/dx)²)^1/2 dx

over the appropriate limits.

So y=lnx would be

ò(1+1/x²)^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 q and f, 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 Þ ds²=dx²+dy²=dx²(1+(dy/dx)²). So since the length of the whole curve is òds (with appropriate limits), this is just ò(1+(dy/dx)²)^1/2 dx, with the corresponding limits.