Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + 4 \cos t \mathbf { j } + 4 \sin t \mathbf { k } , 0 \leq t \leq 2 \pi$
Select the correct answer.
A) $2 \sqrt { 5 } \pi$
B) $4 \sqrt { 5 } \pi$
C) $2 \sqrt { 6 } \pi$
D) $\sqrt { 6 } \pi$

Question

Quizplus · Accepted Answer

The length of a curve given by $\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k}$ where $a \leq t \leq b$ is given by 
$$L = \int_a^b \sqrt{\left(\frac{dx}{dt}ight)^2 + \left(\frac{dy}{dt}ight)^2 + \left(\frac{dz}{dt}ight)^2} dt.$$
Using this formula, we have 
\begin{align*}
L &= \int_0^{2\pi}\sqrt{(2)^2 + (-4\sin t)^2 + (4\cos t)^2} dt \
&= \int_0^{2\pi}\sqrt{16} dt = 4\sqrt{5}\pi.
\end{align*} Therefore, the answer is $\boxed{	ext{(B)}}$.

Find the Length of the Curve r(t)=2ti+4cos⁡tj+4sin⁡tk,0≤t≤2π\mathbf { r } ( t ) = 2 t \mathbf { i } + 4 \cos t \mathbf { j } + 4 \sin t \mathbf { k } , 0 \leq t \leq 2 \pir(t)=2ti+4costj+4sintk,0≤t≤2π

Find the Length of the Curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + 4 \cos t \mathbf { j } + 4 \sin t \mathbf { k } , 0 \leq t \leq 2 \pi$