Chapter 11.3, Problem 10QR

To determine

To find: the length of the curve from $t=0$ to $t=\pi$ when $x=3\cos t$ and $y=5\sin t$ for $0\le t\le 2\pi$ .

Answer to Problem 10QR

The length of the curve is 12.76.

Explanation of Solution

Given information:

  $\begin{array}{l}x=3\cos t\\ y=5\sin t\end{array}$

Time interval $\left(t\right)=0\text{to}\pi$

Formula used:

Length of a curve from the point $x_1\text{to}{x}_2$ is:

  $d={\displaystyle \underset{x_1}{\overset{x_2}{\int }}\left(\sqrt{{\left(du/dx\right)}^2+{\left(dv/dx\right)}^2}\right)}dx$

Where $\left(u,v\right)$ is a point on the curve.

The trigonometric entity ${\sin }^2\theta +{\cos }^2\theta =1$ .

Calculation:

Given that theexpression of $x$ is $3\cos t$ for $0\le t\le 2\pi$ .

Take differentiation of $x$ with respect to $t$ .

  $\begin{array}{c}dx/dt=\frac{d}{dt}3\cos t\\ =-3\sin t\end{array}$

Similarly, the given expression of $y$ is $5\sin t$ for $0\le t\le 2\pi$ .

Take differentiation of $y$ with respect to $t$ .

  $\begin{array}{c}dy/dt=\frac{d}{dt}5\sin t\\ =5\cos t\end{array}$

Now, calculate the length, use the formula $d={\displaystyle \underset{x_1}{\overset{x_2}{\int }}\left(\sqrt{{\left(du/dx\right)}^2+{\left(dv/dx\right)}^2}\right)}dx$ .

  $\begin{array}{c}d={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(\sqrt{{\left(dx/dt\right)}^2+{\left(dy/dt\right)}^2}\right)}dt\\ ={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(\sqrt{{\left(-3\sin t\right)}^2+{\left(5\cos t\right)}^2}\right)}dt\\ ={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(\sqrt{9{\sin }^{2}t+25{\cos }^{2}t}\right)}dt\end{array}$

Substitute $1-{\sin }^{2}t$ for ${\cos }^{2}t$ , we get

  $\begin{array}{c}={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(\sqrt{9{\sin }^{2}t+25\left(1-{\sin }^{2}t\right)}\right)}dt\\ ={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(\sqrt{9{\sin }^{2}t+25-25{\sin }^{2}t}\right)}dt\\ ={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(\sqrt{25-16{\sin }^{2}t}\right)}dt\\ \approx 12.76\end{array}$

Conclusion:

Hence, the required length of the curve is 12.76.