Encontrar la longitud de la curva γ (t)=cos t, sin t, In (cos t ), 0≤t≤π/4
Funciones vectoriales
Encontrar la longitud de la curva
$$\begin{align}&γ(t)=\cos(t), \sin(t), \in (\cos (t) ), 0≤t≤π/4\end{align}$$
Respuesta de Carlos Antonio Gálvez Reyes
3
L=
[Raiz(dX^2+dY^2+dZ^2)]dt .........(1) 0 ≤ t ≤ π/4
X(t)=Cost dX= - Sent
Y(t)=Sent dY= + Cost
Z(t)=Ln(Cos(t)) dZ= - Sent/Cost
L=[Raiz(Sent^2+Cost^2+Sent^2/Cost^2)]dt
L=[Raiz(1+Sent^2/Cost^2)]dt
L=[Raiz(1+Sent^2/Cost^2)]dt
L=[Raiz(1/Cost^2)]dt
L=[(1/Cost)]dt
L=sec t dt = ln|sec t + tan t| 0 ≤ t ≤ π/4
L = Ln|sec π/4 + tan π/4| - Ln|sec 0 + tan 0|
L = Ln|√2 + 1| - Ln|1 + 0|
L = Ln|√2 + 1| - Ln|1|
L = Ln|√2 + 1| - 0
L = Ln(√2 + 1)
L = 0.881373587018
