Question

Find the length of the curve z=z(t)=5e^3it

Answer 1

Given : equation pf the curve is,

$z(t)=x(t)+iy(t)=5e^{3it}=5(\cos 3t+i\sin 3t)\hfill (1)$

On compairing both side we get,

$x=5\cos 3t, y=5\sin 3t$

To find : arc length of (1), of which period of t is $\frac{2\pi}{3}$ we know arc length in (a,b) if functions x(t) and y(t) is,

$L=\int_{a}^{b}\sqrt{(\frac{dx(t)}{dt})^2+(\frac{dy(t)}{dt})^2}$

Step-by-step explanation:

Hence, to find arc length over a half period of (1),

$L=\int_{0}^{\frac{\pi}{3}}\sqrt{(-\frac{5}{3}\sin 3t)^2+(\frac{5}{3}\cos 3t)^2}$

$=\frac{5}{3}\int_{0}^{\frac{\pi}{3}}dt$

$=\frac{5}{3}\times \frac{\pi}{3}=\frac{5\pi}{6}$