Find the length of the curve over the given interval.

Question

Find the length of the curve over the given interval. ​  ​

Quizplus · Accepted Answer

The given interval can be represented as $0\leq t\leq 2\pi$. Using the formula for the length of a curve in polar coordinates, we have:
$$\int_0^{2\pi}\sqrt{r^2+t'^2}~dt=\int_0^{2\pi}\sqrt{4\cos^2(t)+9\sin^2(t)}~dt$$
Using the trigonometric identity $\cos^2(t)+\sin^2(t)=1$, we can simplify the integral as:
$$\int_0^{2\pi}\sqrt{4\cos^2(t)+9(1-\cos^2(t))}~dt=\int_0^{2\pi}\sqrt{9-5\cos^2(t)}~dt$$
This integral can be evaluated using the substitution $u=\sin(t)$ and then using the integral table:
$$\int_0^{2\pi}\sqrt{9-5\cos^2(t)}~dt=\int_0^0\sqrt{9-5u^2}\cos(t)~dt=\left[\frac{9}{5}\sin^{-1}\left(\frac{5}{3}uight)-\frac{9}{5}\left(\frac{5}{3}uight)\sqrt{1-\left(\frac{5}{3}uight)^2}ight]_0^0=0$$
Therefore, the length of the curve over the given interval is 0, and the correct answer is $\boxed{	extbf{(C)}}$.

Find the Length of the Curve Over the Given Interval