Question
Find the arc length of the curve which is parametrized in three dimensions by x(t) = e^t cost , y(t) = e^t sint , z(t) = e^t where 0<=t<=2Pi.
Answers
oobleck
as always, ds = √((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) dt
= √((e^t(cost-sint))^2 + (e^t(sint+cost))^2 + (e^t)^2) dt
= e^t √((cost-sint)^2 + (cost+sint)^2 + 1) dt
= e^t √3 dt
so, the arc length is
∫[0,2π] √3 e^t dt = √3(e^(2π)-1)
= √((e^t(cost-sint))^2 + (e^t(sint+cost))^2 + (e^t)^2) dt
= e^t √((cost-sint)^2 + (cost+sint)^2 + 1) dt
= e^t √3 dt
so, the arc length is
∫[0,2π] √3 e^t dt = √3(e^(2π)-1)