Question

Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22–27.

Short answer

The length of the curve.

Step-by-step solution

Step 1. Given information.

The given curve is

Step 2. Arc length.

The arc-length is given by,

$$\begin{gathered} l(a,b)={\int }_a^b||r^{\text{'}}\left(t\right)||dt \\ ||r^{\text{'}}\left(t\right)||=\sqrt{{\left(e^t\cos t+e^t\sin t\right)}^2+{\left(-e^t\sin t+e^t\cos t\right)}^2+{\left(e^t\right)}^2} \\ =\sqrt{{\left(e^t\right)}^2\left({\cos }^{2}t+{\sin }^{2}t+2\sin t\cos t+{\sin }^{2}t+{\cos }^{2}t-2\sin t\cos t+1\right)} \\ =e^t\sqrt{3} \\ \left(l\right)={\int }_a^b||r^{\text{'}}\left(t\right)||dt \\ ={\int }_0^{\pi }||r^{-1}\left(t\right)||dt \\ {\left.=\sqrt{3}{e}^t\right]}_0^{\pi } \\ =\sqrt{3}\left(e^n-e^0\right) \\ =\sqrt{3}\left(e^n-e^0\right) \end{gathered}$$