Sorry, latex is being weird.(adsbygoogle = window.adsbygoogle || []).push({});

I'm currently trying to come up with a way to find an equation that satisfies:

[tex]s=\int_a^b \sqrt{(f'[x])^2+1} \, dx[/tex]

Which is arc length

and

[tex]G=\int_a^b f[x] \, dx[/tex]

which is area under the curve

where A and s are known values, and f[a]=A, f=B

I've tried expressing f[x] as a power series, namely:

[tex]f[x]=\sum _{n=0}^N a_nx^n[/tex]

so that:

[tex]f'[x]=\sum _{n=1}^N n x^{-1+n} a_n[/tex]

But using that in

[tex]s=\int_a^b \sqrt{(f'[x])^2+1} \, dx[/tex]

has proven to be difficult as I am unclear how to square a power series, let alone how to integrate it's root.

I've also tried adapting the power series into a finite product, namely:

[tex]\text{Log}\left[\sum _{n=0}^N e^{\left(a_nx^n\right)}\right][/tex]

Where f'[x] is:

[tex]f'[x]=\text{Log}\left[\sum _{n=1}^N e^{\left(n a_nx^{n-1}\right)}\right][/tex]

But I encounter a similar problem, for some r, ln[r]^2 can not be simplified.

any insight would be greatly appreciated, thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Determining a function with knowledge of it's arc length and it's definite integral

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**