Find the arc length of the curve defined by $\vec { r } ( t ) = \langle 3 + 4 \cos ( 2 t ) , 4 + 2 \sin ( 4 t ) \rangle$ over $[ 2,4 ]$
A) 4
B) 8
C) 16
D) 32

Question

Quizplus · Accepted Answer

The arc length of a parametric curve $\vec{r}(t) = \langle x(t), y(t) \rangle$ from $t = a$ to $t = b$ is given by $\int_{a}^{b} \sqrt{(x'(t))^2 + (y'(t))^2} \, dt$. For $\vec{r}(t) = \langle 3 + 4 \cos(2t), 4 + 2 \sin(4t) \rangle$, $x'(t) = -8 \sin(2t)$ and $y'(t) = 8 \cos(4t)$. The integral becomes $\int_{2}^{4} \sqrt{(-8 \sin(2t))^2 + (8 \cos(4t))^2} \, dt$, which evaluates to 16.

Find the Arc Length of the Curve Defined By r⃗(t)=⟨3+4cos⁡(2t),4+2sin⁡(4t)⟩\vec { r } ( t ) = \langle 3 + 4 \cos ( 2 t ) , 4 + 2 \sin ( 4 t ) \rangler(t)=⟨3+4cos(2t),4+2sin(4t)⟩

Find the Arc Length of the Curve Defined By $\vec { r } ( t ) = \langle 3 + 4 \cos ( 2 t ) , 4 + 2 \sin ( 4 t ) \rangle$