Chapter 11.2, Problem 61E

Solution

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

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

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

Explanation:

Consider the given statement.

  ${\displaystyle {\int }_a^{b}f\left(x\right)\cdot g\left(x\right)dx=}{\displaystyle {\int }_a^{b}f\left(x\right)dx}\cdot {\displaystyle {\int }_a^{b}g\left(x\right)dx}$

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 ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ .

In the given statement, the left side represents an area but the right side shows product of areas. So, it can be concluded that the left side and right side is not equal in the statement.

Thus, the given statement is false.