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

## Main Question or Discussion Point

Sorry, latex is being weird.

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

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

Which is arc length

and

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

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:

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

so that:

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

But using that in

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

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:

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

Where f'[x] is:

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

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

any insight would be greatly appreciated, thanks!