Chapter 10.1, Problem 32E

To determine

To find: The length of the curve.

Answer to Problem 32E

The answer:

  $L=10$

Explanation of Solution

Given information:

The equations are:

  $\begin{array}{c}x=\frac{{(8t+8)}^{3/2}}{12},\\ y=t^2+t,0\le t\le 2\end{array}$

Calculation:

Know that,

If $dx/dt$

And

  $dy/dt$ are continuous functions of $t$ for $t_1\le t\le t_2$ .then the length of a parametric curve that is traced only once for each $t$ between $t_1$ and $t_2$

equal to:

  

Since,

  $\begin{array}{c}x={(}^8/12,\\ y=t^2+t\text{,}\end{array}$

And

  $0\le t\le 2$

Then,

  $\begin{array}{c}{t}_1=0\\ t_2=2\end{array}$

And,

  $\begin{array}{c}\frac{dx}{dt}=\frac{3}{2}\cdot \frac{{(8t+8)}^{1/2}}{12}\cdot 8\\ =\sqrt{8t+8}\\ \frac{dy}{dt}=2t+1\end{array}$

The foregoing findings lead to the conclusion that the parametric curve is traced only once for each $t$ between $0$ and $2$ that $dx/dt$ and $dy/dt$ are continuous functions of $t$ for $0\le t\le 2$. utilise the above findings to determine that the length of the parametric curve is equal to:

  

Therefore, the required length of the curve is $L=10$ .