Chapter 11.2, Problem 63E

Solution

To check: Whether the given statement is true or false.

The given statement ${\displaystyle {\int }_a^{b}f\left(x\right)}={\displaystyle {\int }_b^{a}f\left(x\right)}$ is false.

Given information: The given statement is “ ${\displaystyle {\int }_a^{b}f\left(x\right)}={\displaystyle {\int }_b^{a}f\left(x\right)}$ .”

Explanation:

Consider the given statement.

  ${\displaystyle {\int }_a^{b}f\left(x\right)}={\displaystyle {\int }_b^{a}f\left(x\right)}$

Definite integral is used to find the area under a curve $f\left(x\right)$ over the interval $\left[a,b\right]$ .

The area under the curve $f\left(x\right)$ over the interval $\left[a,b\right]$ is geometrically represented by ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ .

The area under a curve between two points is considered from left to right.

In the given statement, the area is considered from left to right on the left side. But on the right side the area is considered from right to left that is $b$ to $a$ . So, the right-side expression should be negative.

Thus, the given statement is false.