Chapter 5.2, Problem 50E

To determine

To find: The interval for the integral function ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$ lies between absolute minimum (m) and maximum value (M).

To identify: The property of the integral used to obtain the solution.

Answer to Problem 50E

The value of the integral function lies between $2m$ and $2M$.

The “comparison property 8 of integral” is used to calculate the value of integral.

Explanation of Solution

Given information:

The integral function is ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$.

Show comparison property 8 of integrals:

If $m\le f\left(x\right)\le M$ for $a\le x\le b$, then $m\left(b-a\right)\le {\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}\le M\left(b-a\right)$ (1)

Calculate:

Consider the function $f\left(x\right)$.

The function $f\left(x\right)$ has lower limit $a=0$ and upper limit $b=2$.

The integral function has absolute maximum value as M and absolute minimum value as m.

Modify the function $f\left(x\right)$ as shown below:

$m\le f\left(x\right)\le M\text{for}0\le x\le 2$ (2)

Compare Equation (2) with Equation (1).

The value of the function lies between its absolute maximum and absolute minimum value within limits $\left[0,2\right]$.

Thus, the property of integral “comparison property 8” is satisfied.

Therefore, evaluate the integral ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$ using “comparison Property 8 of integral”.

The expression to find the value of the integral by using Property 8 of integral as shown below:

Substitute 0 for $a$ and 2 for $b$ in Equation (1).

$\begin{array}{l}m\left(2-0\right)\le {\displaystyle {\int }_0^{2}f\left(x\right)dx}\le M\left(2-0\right)\\ 2m\le {\displaystyle {\int }_0^{2}f\left(x\right)dx}\le 2M\end{array}$

Conclusion:

Compare Equation (2) with Equation (1).

The value of the function lies between its absolute maximum and absolute minimum value within limits $\left[0,2\right]$.

Thus, the condition to apply comparison property 8 of integral is satisfied.

Therefore, the “comparison Property 8 of integral” is used to evaluate function ${\displaystyle {\int }_0^{2}f\left(x\right)dx}$.