Verified answer

The arc length is

∫ (0 to 1) √{[x'(t)]² + [y'(t)]² + [z'(t)]²} dt

where x'(t), y'(t), and z'(t) are the respective components of r(t). Therefore

x'(t) = 8√2

y'(t) = 8e^(4t)

z'(t) = -8e^(-4t)

which gives us

∫ (0 to 1) √{(8√2)² + [8e^(4t)]² + [-8e^(-4t)]²}dt =

∫ (0 to 1) √[(64)(2) + 64e^(8t) + 64e^(-8t)]dt =

8∫ (0 to 1) √[e^(8t) + 2 + e^(-8t)]dt =

8∫ (0 to 1) √{[e^(4t)]² + 2[e^(4t)][e^(-4t)] + [e^(-4t)]²}dt =

8∫ (0 to 1) √[e^(4t) + e^(-4t)]²dt =

8∫ (0 to 1) [e^(4t) + e^(-4t)]dt =

8[(1/4)e^(4t) - (1/4)e^(-4t)] (0 to 1) =

8[(1/4)e^4 - (1/4)e^(-4) - 1/4 + 1/4] =

2e^4 - 2e^(-4).

I hope that helps!