Find the length of the curve $\mathbf { r } ( t ) = 8 t \mathbf { i } + 3 \cos t \mathbf { j } + 3 \sin t \mathbf { k }$ $0 \leq t \leq 2 \pi$
A) $2 \sqrt { 73 }$ $\pi$
B) $\sqrt { 73 }$ $\pi$
C) $\sqrt { 11 }$ $\pi$
D) $2 \sqrt { 11 }$ $\pi$

Question

Quizplus · Accepted Answer

The length of the curve is found by integrating the magnitude of the derivative of $\mathbf{r}(t)$. The derivative is $\mathbf{r}'(t) = 8\mathbf{i} - 3\sin t \mathbf{j} + 3\cos t \mathbf{k}$. The magnitude is $\sqrt{8^2 + (-3\sin t)^2 + (3\cos t)^2} = \sqrt{64 + 9} = \sqrt{73}$. Integrating from $0$ to $2\pi$ gives $2\sqrt{73}\pi$.

Find the Length of the Curve r(t)=8ti+3cos⁡tj+3sin⁡tk\mathbf { r } ( t ) = 8 t \mathbf { i } + 3 \cos t \mathbf { j } + 3 \sin t \mathbf { k }r(t)=8ti+3costj+3sintk

Find the Length of the Curve $\mathbf { r } ( t ) = 8 t \mathbf { i } + 3 \cos t \mathbf { j } + 3 \sin t \mathbf { k }$