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 is found by integrating the magnitude of the derivative of $\mathbf{r}(t)$. The derivative is $\mathbf{r}'(t) = -2\mathbf{i} - \mathbf{j} + \mathbf{k}$. The magnitude is $\sqrt{(-2)^2 + (-1)^2 + 1^2} = \sqrt{6}$. Integrating from $-2$ to $1$ gives $(1 - (-2)) \times \sqrt{6} = 3\sqrt{6}$.

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 $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1$