Chapter 11.2, Problem 62E

Solution

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

The given statement ${\displaystyle {\int }_a^{c}f\left(x\right)dx}+{\displaystyle {\int }_c^{b}f\left(x\right)dx}={\displaystyle {\int }_a^{b}f\left(x\right)dx}$ is true.

Given information: The given statement is “ ${\displaystyle {\int }_a^{c}f\left(x\right)dx}+{\displaystyle {\int }_c^{b}f\left(x\right)dx}={\displaystyle {\int }_a^{b}f\left(x\right)dx}$ for $a<c<b$ .”

Explanation:

Consider the given statement.

  ${\displaystyle {\int }_a^{c}f\left(x\right)dx}+{\displaystyle {\int }_c^{b}f\left(x\right)dx}={\displaystyle {\int }_a^{b}f\left(x\right)dx}$

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 the sum of areas under the curve $f\left(x\right)$ over the intervals $\left[a,c\right]$ and $\left[c,b\right]$ . It is given that $a<c<b$ , so the right side represents the same area under the curve $f\left(x\right)$ over the interval $\left[a,b\right]$ . So, the right side is equal to the left side.

Thus, the given statement is true.