3. Find the length of the curve given by r(t)= √(2)ti +e^t j +e^-t k, 0≤t≤1.

Answer 1

ds^2 = t^2 + e^2t + e^-2t = (e^t + e^-t)^2
ds = e^t + e^-t = 2cosh(t)
so the arc length is
∫[0,1] 2cosh(t) dt = 2sinh(t) [0,1] = 2sinh(1) = e - 1/e

Answer 2

ds = e^t + e^-t = 2cosh(t) what this mean? does it mean e^t +e^-t is the same ascosh(t)?