Chapter 6.2, Problem 37E

(a)

To determine

The points of discontinuity of the integrand on the interval of integration.

Answer to Problem 37E

The Function is not continuous at $x=0$ .

Explanation of Solution

Given information:

The given function is,

  ${\displaystyle \underset{-2}{\overset{3}{\int }}\frac{x}{\left|x\right|}}dx$

Formula used:

  $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)=f\left(a\right)$

Calculation:

For f(x) to be continuous a point x = a it must be satisfied this condition

  $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)=f\left(a\right)$

Now,

  $f\left(x\right)=\frac{x}{\left|x\right|}$

  $\left|x\right|$ is equal to x if x is greater than zero and equal to -x if x is less than zero.

So, the check continuity at

  $\begin{array}{l}x=0\\ \underset{x\to 0-}{\lim }\frac{x}{\left|x\right|}=\frac{x}{-x}\\ =-1\\ \underset{x\to 0-}{\lim }\frac{x}{\left|x\right|}=\frac{x}{x}\\ =1\end{array}$

Hence,

  $\underset{x\to 0-}{\lim }\frac{x}{\left|x\right|}\ne \underset{x\to 0+}{\lim }\frac{x}{\left|x\right|}$

Function is not continuous at

  $x=0$

Conclusion:

The Function is not continuous at $x=0$

(b)

To determine

The area of the integral.

Answer to Problem 37E

The value is ${\displaystyle \underset{-2}{\overset{3}{\int }}\frac{x}{\left|x\right|}}dx=1$

Explanation of Solution

Given information: The function is

  ${\displaystyle \underset{-2}{\overset{3}{\int }}\frac{x}{\left|x\right|}}dx$

Formula used:

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=\left(\text{area above the x-axis}\right)-\left(\text{area below the x-axis}\right)$

Calculation:

For any integral function,

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=\left(\text{area above the x-axis}\right)-\left(\text{area below the x-axis}\right)$

  

So,

  $\begin{array}{l}{\displaystyle \underset{-2}{\overset{3}{\int }}\frac{x}{\left|x\right|}}dx=\left(\text{area of rectangle above the x-axis}\right)-\left(\text{area of rectangle below the x-axis}\right)\\ =\left(3\times 1\right)-\left(2\times 1\right)\\ =3-2\\ {\displaystyle \underset{-2}{\overset{3}{\int }}\frac{x}{\left|x\right|}}dx=1\end{array}$

Conclusion:

The value is ${\displaystyle \underset{-2}{\overset{3}{\int }}\frac{x}{\left|x\right|}}dx=1$