Chapter 7, Problem 10PS

Using a Function Let f be rectifiable on the interval [a, b], and let

$s\left(x\right)={\displaystyle {\int }_a^x{\sqrt{1+\left[f^{\prime }\left(t\right)\right]}}^2}dt.$

(a) Find $\frac{ds}{dx}$.

(b) Find ds and ${\left(ds\right)}^2$.

(c) Find $s\left(x\right)$ on [1, 3] when $f\left(t\right)=t^{3/2}$

(d) Use the function and interval in part (c) to calculate s(2) and describe what it signifies.