Distance Traveled by A Particle | Parametric Equations, AP Calculus BC

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Let x(t)=<rcost, rsint, r> where r, the radius of the helix path that the particle travels, varies with time. v(t) = x'(t) = <r'cost-rsint, r'sint+rcost, r'> Since speed is the magnitude of velocity, we have |v(t)| = After expanding the brackets and simplifying, |v(t)| = sqrt(2(r')^2+r^2). Plugging in r = e^t and integrating with respect to time yields a total distance of sqrt3*(e^4-1).