Chapter 6, Problem 55RE

(a)

To determine

To calculate:The value of the number on the calculator.

Answer to Problem 55RE

The value is ${\displaystyle \underset{-20}{\overset{20}{\int }}\left(e^{\frac{-x^2}{2}}\right)}dx=2.506628275$

Explanation of Solution

Given information:

The graph approaches the x-axis as an asymptote both to the left and the right, but quickly enough so that the total area is a finite number. In fact,

  $\text{NINT}\left(e^{-x^2/2},x,-10,10\right)$

Formula used:

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=NINT\left(f\left(x\right),x,a,b\right)$

Calculation:

(a) Calculator has a numerical integration capability, which will denote as NINT. In particular, use NINT (f (x), x, a, b) to denote a calculator approximation

or ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$ when write

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=NINT\left(f\left(x\right),x,a,b\right)$

The understanding that the right-hand side of the equation is an approximation of the left-hand side

Now, use above expression with this function

  $\begin{array}{l}f\left(x\right)=e^{\frac{-x^2}{2}}\\ {\displaystyle \underset{-10}{\overset{10}{\int }}\left(e^{\frac{-x^2}{2}}\right)}dx=NINT\left(e^{\frac{-x^2}{2}},x,-10,10\right)\\ {\displaystyle \underset{-10}{\overset{10}{\int }}\left(e^{\frac{-x^2}{2}}\right)}dx=2.506628275\end{array}$

Now,

  $\begin{array}{l}{\displaystyle \underset{-20}{\overset{20}{\int }}\left(e^{\frac{-x^2}{2}}\right)}dx=NINT\left(e^{\frac{-x^2}{2}},x,-20,20\right)\\ {\displaystyle \underset{-20}{\overset{20}{\int }}\left(e^{\frac{-x^2}{2}}\right)}dx=2.506628275\end{array}$

Conclusion:

The value is ${\displaystyle \underset{-20}{\overset{20}{\int }}\left(e^{\frac{-x^2}{2}}\right)}dx=2.506628275$

(b)

To determine

To find: The value of the area.

Answer to Problem 55RE

The value is $\sqrt{2\pi }=2.506628275$

Explanation of Solution

Given information:

The graph approaches the x-axis as an asymptote both to the left and the right, but quickly enough so that the total area is a finite number. In fact,

  $\text{NINT}\left(e^{-x^2/2},x,-10,10\right)$

Formula used:

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=NINT\left(f\left(x\right),x,a,b\right)$

Calculation:

  $\begin{array}{l}\pi =3.14159264\\ 2\pi =6.283185275\\ \sqrt{2\pi }=2.506628275\end{array}$

Conclusion:

The value is $\sqrt{2\pi }=2.506628275$