Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

• the signed area between the graph of a function f and the x-axis on $[a,b]$
  

The signed area between the graph function $f\left(x\right)$ and x-axis on$[a,b]$.

X-axis on an interval $[a,b]$, i.e., $A\left(\text{area}\right)={\int }_a^{b}f\left(x\right)dx$.

The function $f\left(x\right)$may be above or below the x-axis, although the area is always a positive quantity, it can bear a sign ' + ' or '-' according to

(i). If $f\left(x\right)\ge 0$on all the interval $[a,b]$then $A\left(\text{area}\right)={\int }_a^{b}f\left(x\right)dx\ge 0$, i.e. A is positive.

(ii). If $f\left(x\right)\le 0$ on all the interval $[b,c]$ then $A\left(\text{area}\right)={\int }_b^{c}f\left(x\right)dx\le 0$, i.e. A is negative.

The signed areas (positive and negative) are shown in the figure here.

Since, the area $\mathrm{A}$between the curve and x-axis can have positive and negative values, however negative are in actual calculations has no meaning or sometimes maybe absurd, e.g. Area of the square is $-25\mathrm{sq}.\mathrm{cm}$then its side will be $\sqrt{-25}=\pm 5i$an imaginary value. Therefore, to avoid such an absurdity the computed are is $takenA=\left|{\int }_b^{c}f\left(x\right)dx\right|$.