91Ó°ÊÓ

$\text{The objective is to construct an example for a function}f\text{for which the signed area between}f$

$\text{and the}x\text{-axis on}[0,4]\text{is}0\text{also another function}g\text{for which the absolute area between}g$

$\text{and the}x\text{-axis on}[0,4]\text{is}0\text{.}$

$\text{The example for signed area is,}$

$\begin{array}{r}{∫}_0^4 2x−4dx\\ ={\left[\frac{2x^2}{2}−4x\right]}_0^4\\ ={\left[x^2−4x\right]}_0^4\\ =16−16\\ =0\end{array}$

So,$f\left(x\right)=2x-4$

$\text{The example for absolute area is,}$

$\begin{array}{r}\frac{1}{4−0}{∫}_0^4 4x−8dx\\ =\frac{1}{4}{\left[\frac{4x^2}{2}−8x\right]}_0^4\\ =\frac{1}{4}{\left[2x^2−8x\right]}_0^4\\ =\frac{1}{4}[32−32]\\ =0\end{array}$

So, $g\left(x\right)=4x-8$

$\text{Therefore, the examples are}f\left(x\right)=2x−4\text{and}g\left(x\right)=4x−8$

$\text{The objective is to construct an example for a function}f\text{for which the signed area between}f$

$\text{and the}x\text{-axis on}[0,5]\text{is less than the signed area on}[0,3]\text{.}$

$\text{The example is}f\left(x\right)=1$

$\text{The signed area on}[0,5]\text{is,}$

$\begin{array}{r}{∫}_0^5 1dx\\ =[x{]}_0^5\\ =5\end{array}$

$\text{And, signed area on}[0,3]\text{is}$

$\begin{array}{r}{∫}_0^3 1dx\\ =[x{]}_0^3\\ =3\end{array}$

$\text{Therefore, the example is}f\left(x\right)=1\text{.}$