Chapter 7.4, Problem 19E

To determine

To find: $\text{(a)}$ the curve through the point $(1,1)$ .

  $(b)$ The numbers of curves are there.

Answer to Problem 19E

  $\text{(a)}$ The curve is $y=\sqrt{x}$ .

  $(b)$ Only one curve is there.

Explanation of Solution

Given information:

The length of integral is

Calculation:

For

The given integral is

This implies that,

  $\begin{array}{c}{\left(\frac{dy}{dx}\right)}^2=\frac{1}{4x}\\ \Rightarrow \frac{dy}{dx}=\sqrt{\frac{1}{4x}}\\ =\frac{1}{2\sqrt{x}}\end{array}$

Integrate both sides of the equation and evaluate it to get the curve's equation.

  

Let find the value of $\text{ C}$ use the given point $(1,1)$ :

  $\begin{array}{c}y=\sqrt{x}+C\\ \Rightarrow 1=\sqrt{1}+C\\ \Rightarrow C=1-\sqrt{1}\\ \Rightarrow C=1-1\\ \Rightarrow C=0\end{array}$

Since, the equation of the curve is:

  $\begin{array}{c}y=\sqrt{x}+C\\ \Rightarrow y=\sqrt{x}+0\\ \Rightarrow y=\sqrt{x}\end{array}$

And

For $(b)$ Let's go back at item to see how many curves there are $(a)$ . Be aware that there is only one point through the curve provided. If so, it follows that the derivative of the function and its value at a certain value of $x$ are also one. As a result, only one curve may be drawn from the specified length integral.

Therefore, the required $\text{(a)}$ the curve through the point $(1,1)$ is $y=\sqrt{x}$ .

  $(b)$ Only one curve is there.