Chapter 6.4, Problem 33ES

(a) Show that the total arc length of the ellipse

$x=2\cos t,y=\sin t\left(0\le t\le 2\pi \right)$

is given by

${\displaystyle {\int }_0^{\pi /2}\sqrt{1+3{\sin }^{2}tdt}}$

(b) Use a CAS or a scientific calculator with a numerical integration capability to approximate the arc length in part (a). Round your answer to two decimal places.

(c) Suppose that the parametric equations in part (a) describe the path of a particle moving in the $xy\text{-plane}$ , where $t$ is time in seconds and $x$ and $y$ are in centimeters. Use a CAS or a scientific calculator with a numerical integration capability to approximate the distance traveled by the particle from $t=1.5\text{ s}$ to $t=4.8\text{ s}$ . Round your answer to two decimal places.