2. Let C be the curve in R³ defined by the parametric equations

x(t) = cos(e^t)

y(t) = sin(e^t)

z(t) = e^t

for t in [0,2]. What is the length of C?

The length of a curve C in R³ is the integral of the square root of the sum of the squares of the derivatives of each parametric curve over the interval [a,b]. Thus, the derivatives of the above parametric equations are,

So that the length of curve C, denoted L, is