Chapter 11.1, Problem 30E

To determine

To estimate: the area of the region bounded by the $x-axis$ .

Answer to Problem 30E

The area of the region bounded by $x-axis$ is $0.997units$ .

Explanation of Solution

Given:

  $f\left(x\right)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$

the vertical lines $x=-3$ and $x=3$ .

Calculation:

First, sketch the graph of the standard normal distribution function,

  $f\left(x\right)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}$

For this use that the normal probability distribution function is given by

  $f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}$

Compare this function with the given function to find values of mean and standard deviation.

Thus, $\sigma =1$ and $\mu =0$

Sketch a graph with mean $0$ and standard deviation $1$ and scale the $x-axis$ with markings at

  $\mu \pm \sigma ,\mu \pm 2\sigma$ and $\mu \pm 3\sigma$

Thus, the graph of the given function like

  

To estimate the area of the region bounded by $x-axis$ , the graph of $f$ , and the vertical lines $x=3$ and $x=-3$ ,

Since, the graph is a normal curve and the values $3$ and $-3$ represents three standard deviations to either side of the mean. Then using the property of normal distribution that about $99.7\%$ of the area lies within $3$ standard deviations of mean, it can be said that

Since, the area under the curve is equal to $1$ . Hence, the area of the region bounded by the $x-axis$ , the graph of $f$ , and the vertical lines $x=3$ and $x=-3$ is $0.997units$ .

Conclusion:

Hence, the area of the region bounded by $x-axis$ is $0.997units$ .