Chapter 11.1, Problem 43E

To determine

To calculate: The length of one arc of the curtate cycloid defined paramerically.

Answer to Problem 43E

The required length of the arc is 21.01.

Explanation of Solution

Given information:

The parametric equations: $x=3t-2\sin t$ , $y=3-2\cos t$

Calculation:

The given parametric equations are $x=3t-2\sin t$ and $y=3-2\cos t$

Differentiate the parametric equations with respect to t .

  $\frac{dx}{dt}=3-2\cos t$ , $\frac{dy}{dt}=2\sin t$

Consider that L be the length of the parametric curve.

So, $L={\displaystyle \underset{t_2}{\overset{t_2}{\int }}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}}dt$

Write the length of one arc as:

  $\begin{array}{c}L={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{{\left(3-2\cos t\right)}^2+\left(2\sin t\right)}dt}\\ ={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{9-12\cos t+4{\cos }^{2}t+4{\sin }^{2}t}dt}\\ ={\displaystyle \underset{0}{\overset{2\pi }{\int }}\sqrt{13-12\cos t}dt}\\ =21.01\end{array}$

Hence, the required length of the arc is 21.01.