Find the length of the curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1$.
A) $3 \sqrt { 6 }$
B) $\sqrt { 6 }$
C) $2 \sqrt { 6 }$
D) $6 \sqrt { 6 }$

Question

Quizplus · Accepted Answer

The length of the curve $\mathbf{r}(t)$ from $t = a$ to $t = b$ is given by the integral $\int_{a}^{b} \|\mathbf{r}'(t)\| dt$, where $\|\mathbf{r}'(t)\|$ is the magnitude of the derivative of $\mathbf{r}(t)$. For $\mathbf{r}(t) = -2t\mathbf{i} - t\mathbf{j} + t\mathbf{k}$, the derivative $\mathbf{r}'(t) = -2\mathbf{i} - \mathbf{j} + \mathbf{k}$. The magnitude of $\mathbf{r}'(t)$ is $\sqrt{(-2)^2 + (-1)^2 + 1^2} = \sqrt{6}$. The integral from $-2$ to $1$ is $\int_{-2}^{1} \sqrt{6} dt = \sqrt{6}(1 - (-2)) = 3\sqrt{6}$.

Find the Length of the Curve r(t)=−2ti−tj+tk,−2≤t≤1\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1r(t)=−2ti−tj+tk,−2≤t≤1

Find the Length of the Curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1$