Find the length of the curve $\mathbf { r } ( t ) = \left\langle \sin 2 t , \cos 2 t , 2 t ^ { \frac { 3 } { 2 } } \right\rangle 0 \leq t \leq 1$
A) $\frac { 2 } { 27 } ( 13 \sqrt { 13 } - 8 )$

B) $\frac { 13 } { 9 }$
C) $\frac { 13 \sqrt { 13 } - 6 } { 27 }$
D) $\frac { 16 } { 9 }$
E) $\frac { 10 } { 27 } ( \sqrt { 13 } - 2 )$
F) $\frac { 19 } { 9 }$

G) $\frac { 4 } { 9 } ( 7 \sqrt { 7 } - 2 )$

H) $\frac { 22 } { 9 }$

Question

Find the length of the curve $\mathbf { r } ( t ) = \left\langle \sin 2 t , \cos 2 t , 2 t ^ { \frac { 3 } { 2 } } \right\rangle 0 \leq t \leq 1$ 
A) $\frac { 2 } { 27 } ( 13 \sqrt { 13 } - 8 )$

B) $\frac { 13 } { 9 }$ 
C) $\frac { 13 \sqrt { 13 } - 6 } { 27 }$ 
D) $\frac { 16 } { 9 }$ 
E) $\frac { 10 } { 27 } ( \sqrt { 13 } - 2 )$
F) $\frac { 19 } { 9 }$

G) $\frac { 4 } { 9 } ( 7 \sqrt { 7 } - 2 )$

H) $\frac { 22 } { 9 }$

Quizplus · Accepted Answer

The length of the curve $\mathbf{r}(t) = \langle \sin 2t, \cos 2t, 2t^{\frac{3}{2}} \rangle$ from $t = 0$ to $t = 1$ is found using the formula for the arc length of a parametric curve: $L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} dt$. Calculating the derivatives of each component of $\mathbf{r}(t)$ and substituting them into the formula, then integrating from 0 to 1, gives the length as $\frac{2}{27}(13\sqrt{13} - 8)$.

Find the Length of the Curve r(t)=⟨sin⁡2t,cos⁡2t,2t32⟩0≤t≤1\mathbf { r } ( t ) = \left\langle \sin 2 t , \cos 2 t , 2 t ^ { \frac { 3 } { 2 } } \right\rangle 0 \leq t \leq 1r(t)=⟨sin2t,cos2t,2t23​⟩0≤t≤1

Find the Length of the Curve $\mathbf { r } ( t ) = \left\langle \sin 2 t , \cos 2 t , 2 t ^ { \frac { 3 } { 2 } } \right\rangle 0 \leq t \leq 1$