Chapter 12.3, Problem 36ES

Show That in spherical coordinates a curve given by the parametric equation $\rho =\rho \left(t\right),\theta =\theta \left(t\right),\varphi =\varphi \left(t\right)\text{for}a\le t\le b$ has arc length

$L={\displaystyle {\int }_a^b\sqrt{{\left(\frac{d\rho }{dt}\right)}^2+{\rho }^2{\sin }^2\varphi {\left(\frac{d\theta }{dt}\right)}^2+{\rho }^2{\left(\frac{d\varphi }{dt}\right)}^2}dt}$