Chapter 11.1, Problem 38E

To determine

To Show:Thatthe length formula can be reduce to Cartesian formula $L={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}dt$ .

Explanation of Solution

Given Information:

  $x=a\left(t-\sin t\right)$  ........ (1)

  $y=a\left(1-\cos t\right)$  ........ (2)

  $a\le x\le b$

Formula Used:

Length of a curve is given by

  $L={\displaystyle \underset{t1}{\overset{t2}{\int }}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}}dt$  ........ (3)

Calculation:

Replace $t$ with $x$ in the equation (3)

  $\begin{array}{c}L={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}}dt\\ L={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{{\left(\frac{dx}{dx}\right)}^2+{\left(\frac{dy}{dx}\right)}^2}}dt\\ ={\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}}dt\end{array}$

Conclusion:

Hence, the required derivation is shown.