The arc$-$length function $s(t)of$ a given path $c(x)is$ defined $by$:

$s(t)={\displaystyle \underset{a}{\overset{t}{}}}||c^{\text{'}}(x)||dx$

$It$ measures the length $of$ the path $c(x)$ from $x=atox=t.$

$(a)$ Compute the arclength function $s(t)of{c}_1(t)=(cos(t),sin(t),t)$ with $a=0.$

$(b)$ Compute the arclength function $s(t)of{c}_2(t)=(cosh(t),sinh(t),t)$ with $a=0.$

The definitions $ofcosh$ and $sinh$ are $as$ follows:

$cosh(t)=\frac{e^x+e^{-x}}{2},sinh(t)=\frac{e^x-e^{-x}}{2}$

They are related $to$ parametrizing the hyperbola $x^2-y^2=1in$ the same way that $cos$ and $sin$ are related $to$ parametrizing the circle

$x^2+y^2=1.$ Thus they are called the hyperbolic trigonometric functions.