Question

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

12. The area accumulation function for a function $f$on an interval $[a,b]$.

Short answer

The area accumulation function for $f$on $[a,b]$is the function that, for $x\in [a,b]$, is equal to the signed area between the graph of $f$and the $x$-axis on $[a,b]$,

$A\left(x\right)={\int }_a^{x}f\left(t\right)dt$

Step-by-step solution

Step 1. Given data 

We have to define the area accumulation function for a function $f$on an interval$[a,b]$.

Step 2. Definition

Let $f$is a continuous function on $[a,b]$, then the area accumulation function for $f$on $[a,b]$is the function that, for $x\in [a,b]$, is equal to the signed area between the graph of $f$and the $x$-axis on $[a,b]$,

$A\left(x\right)={\int }_a^{x}f\left(t\right)dt$