Find the length of the curve.
-$$x = \cos 2 t , y = \sin 2 t , 0 \leq t \leq 2 \pi$$
A) $6 \pi$
B) $\pi$
C) $4 \pi$
D) $2 \pi$

Question

Quizplus · Accepted Answer

We can use the formula for arc length in parametric form:

$$\int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} dt$$

Here, we have $x = \cos 2t$ and $y=\sin 2t$, so

$$\frac{dx}{dt} = -2\sin 2t$$
$$\frac{dy}{dt} = 2\cos 2t$$

and therefore the arc length is

$$\int_{0}^{2\pi} \sqrt{(-2\sin 2t)^2 + (2\cos 2t)^2} dt$$

Simplifying the expression inside the square root, we get

$$\sqrt{4\sin^2 2t + 4\cos^2 2t} = 2$$

so the arc length is

$$\int_{0}^{2\pi} 2 dt = \boxed{4\pi}.$$

Find the Length of the Curve x=cos⁡2t,y=sin⁡2t,0≤t≤2πx = \cos 2 t , y = \sin 2 t , 0 \leq t \leq 2 \pix=cos2t,y=sin2t,0≤t≤2π