Chapter 5.3, Problem 47E

To determine

To check : The option that is not true..

Answer to Problem 47E

The answer: Option $(A)$ is not true.

Explanation of Solution

Given information:

The integrations are:

  ${{\displaystyle \int }}_3^{7}f(x)dx=5$ and ${{\displaystyle \int }}_3^{7}g(x)dx=3$ .

Calculation:

  $(A)$ The following is NOT true in the general situation, hence the supplied statement may or may not be true:

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

Therefore, the given statement is not true.

  $(B)$ Using the sum rule for integration,

  $\begin{array}{c}{{\displaystyle \int }}_3^{7}f(x)+g(x)dx={{\displaystyle \int }}_3^{7}f(x)dx+{{\displaystyle \int }}_3^{7}g(x)dx\\ =5+3\\ =8\end{array}$

Therefore the statement given in this option is correct.

  $(C)$ Using the sum rule for integration,

  $\begin{array}{c}{{\displaystyle \int }}_3^{7}f(x)-g(x)dx={{\displaystyle \int }}_3^{7}f(x)dx-{{\displaystyle \int }}_3^{7}g(x)dx\\ =5-3\\ =2\end{array}$

Therefore the statement given in this option is correct.

  $(D)$ When reverse the order of the limits, the definite integral gets multiplied by $-1$

  $\begin{array}{c}{{\displaystyle \int }}_7^{3}g(x)-f(x)dx=-{{\displaystyle \int }}_3^{7}g(x)-f(x)dx\\ ={{\displaystyle \int }}_3^{7}f(x)-g(x)dx\end{array}$

This is the same integral that encountered in Option $(C)$ , this is equal to 2 and Option $(D)$ is also correct.