What I would like to know is some general formula for the length of a function’s line between two point $a$ and $b$ or $f(a)$ and $f(b)$.
For example the length of the function $f(x) = x$ is $x \sqrt2$. We can call this new length function $g(x)$ and to find the length of the function $f(x)$ between $a$ and $b$ to be equal to $g(b)-g(a)$.
This of course gets more complicated when you consider functions of degrees other than $1$.
Bonus points for anyone able to generalize this further into mult-variable functions.
My ugly solution
Length from $a$ to $b$ = limit as $n$ goes to infinity the sum from $k=0$ to $n-1 /\sqrt {(f(b-(k((b-a)/n)))-(f(b-((k+1)((b-a)/n)))))^2)+((b-a)/n))^2}$ I’ll understand if you downvote just because of this whole mess.
P.S. Edits are very much appreciated especially with the math sysmbols as well as comments telling me how I can improve this question or make it more clear.
Return answers to this question must show sources or thought process as well as what area of math they used. Thanks for the help.
This is a standard Calculus problem. The length of the graph of $f\colon[a,b]\longrightarrow\mathbb R$ is$$\int_a^b\sqrt{1+\left(f'(t)\right)^2}\,\mathrm dt,$$assuming that $f$ is differentiable and that $f'$ is continuous.
And if $f:[a,b]\times[c,d]\longrightarrow\mathbb{R}$ is differentiable with continuous partial derivatives, you can compute the area if its graph using the formula:$$\int_a^b\int_c^d\sqrt{1+\left(\frac{\partial f}{\partial x}(x,y)\right)^2+\left(\frac{\partial f}{\partial y}(x,y)\right)^2}\,\mathrm dx\,\mathrm dy.$$