Chapter 7.8, Problem 1PT

True or False:

${\displaystyle {\int }_{-2}^3\left|x\right|}dx$ is an improper integral.

To determine

Whether the statement “${\displaystyle {\int }_{-2}^3\left|x\right|dx}$ is an improper integral” is true or false.

Answer to Problem 1PT

The given statement is $\underset{\_}{\text{false}}$.

Explanation of Solution

Given:

The integral is ${\displaystyle {\int }_{-2}^3\left|x\right|dx}$.

Definition used:

The integral ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$ is an Improper integral if:

(1) $a=-\infty$ or $b=\infty \left(\text{or both}\right)$ then, ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$ is an improper integral.

(2) ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$ has a vertical asymptote at either a, b or at some point c between a and b.

Calculation:

Consider the integral ${\displaystyle {\int }_{-2}^3\left|x\right|dx}$.

From the definition above, it can be clearly seen that the integral limits are finite and thus from definition (1) the given integral is not an improper integral.

A vertical asymptote exists if at least one of the following statements is satisfied, that is,

$\begin{array}{l}\underset{x\to a^{-}}{\lim }f\left(x\right)=\pm \infty \\ \underset{x\to a^{+}}{\lim }f\left(x\right)=\pm \infty \end{array}$

Here, the function $\left|x\right|$ has no undefined points.

For the given function both the left hand limit and the right and limit exists and thus, there is no vertical asymptote at either a, b or at some point c between a and b.

Thus, the integral ${\displaystyle {\int }_{-2}^3\left|x\right|dx}$ is not an improper integral.

Therefore, the given statement is $\underset{\_}{\text{false}}$.