The accumulation function and its properties

A continuous function $\displaystyle f$ describes the rate of change of a quantity. The accumulation function of $\displaystyle f$ from $\displaystyle a$, $\displaystyle F_a(x)$, is the quantity accumulated from $\displaystyle a$ to $\displaystyle x$: the area between the graph of $\displaystyle f$ and the $\displaystyle x$-axis from $\displaystyle a$ to $\displaystyle x$, with the area below the axis counted as negative. As $\displaystyle x$ moves right, the area grows or shrinks, so it is a function of $\displaystyle x$.

• The sign $\displaystyle \int$ is a long S, from the word Sum: the accumulation is a limit of sums of rectangles.
• $\displaystyle f(t)\,dt$ recalls a rectangle: a height $\displaystyle f(t)$ times a tiny width $\displaystyle dt$.
• $\displaystyle a$ is the lower limit and $\displaystyle x$ the upper limit.
• The variable inside the integral is called $\displaystyle t$, so as not to mix it up with $\displaystyle x$ – the end up to which it accumulates.
• Its value at a given point, $\displaystyle {F_a(b)}={\int_a^b f(t)\,dt}$, is a number: the definite integral of $\displaystyle f$ from $\displaystyle a$ to $\displaystyle b$.

Read the properties of the accumulation function from the graph of $\displaystyle f$, with no formula:

• $\displaystyle {F_a(a)}={0}$: from $\displaystyle a$ to $\displaystyle a$ nothing has accumulated yet.
• Where $\displaystyle f$ is positive, a positive quantity accumulates, and $\displaystyle F_a$ rises.
• Where $\displaystyle f$ is negative, $\displaystyle F_a$ falls.
• Where $\displaystyle f$ changes sign from + to −, $\displaystyle F_a$ has a maximum; from − to + – a minimum.
• Where $\displaystyle f$ rises, each piece adds more than the one before, and $\displaystyle F_a$ is concave up; where $\displaystyle f$ falls – concave down.
• At an extremum of $\displaystyle f$ the concavity of $\displaystyle F_a$ changes: $\displaystyle F_a$ has an inflection point there.

The example below reads the properties of the distance function from the speed of a toy car: $\displaystyle {f(t)}={4t-t^2}$ metres a second, $\displaystyle 0\le t\le4$, and the distance $\displaystyle {F_0(x)}={\int_0^x f(t)\,dt}$:

• $\displaystyle f>0$ throughout, so the distance $\displaystyle F_0$ rises all the time, from $\displaystyle {F_0(0)}={0}$.
• Until $\displaystyle {t}={2}$ the speed rises: the distance rises faster and faster, concave up.
• After $\displaystyle {t}={2}$ the speed falls: the distance still rises, but more slowly – concave down.
• At $\displaystyle {t}={2}$, the maximum of the speed, the graph of the distance has an inflection point.

This is the graph of $\displaystyle F_0$ (you will find its formula in the next lesson):

Changing the lower limit shifts the graph of the accumulation function up or down, and keeps its shape:

• $\displaystyle {F_b(x)}={F_a(x)-F_a(b)}$: what accumulates from $\displaystyle b$ is what accumulates from $\displaystyle a$, less what accumulated from $\displaystyle a$ to $\displaystyle b$.
• $\displaystyle F_a(b)$ is a constant number, so the difference between the two accumulation functions is constant.
• So the rise, the fall, the extremum and inflection points – at the same $\displaystyle x$; only the height changes.

The board below shows the graph of $\displaystyle f$.
$\displaystyle {F(x)}={\int_0^x f(t)\,dt}$.

Calculate $\displaystyle F(2)$, $\displaystyle F(4)$ and $\displaystyle F(6)$ from the areas of the triangles.

Where does $\displaystyle F$ rise, and where does it fall? What are the extremum points of $\displaystyle F$, and of what kind?

For $\displaystyle 0<x<4$ the function $\displaystyle f$ falls. Which way is $\displaystyle F$ concave there?

The function $\displaystyle {f(x)}={x^2-4x+3}$ is given.
$\displaystyle {G(x)}={\int_0^x f(t)\,dt}$.

Mark on the board the points on the $\displaystyle x$-axis at whose $\displaystyle x$ the function $\displaystyle G$ has extremum points.

Where does $\displaystyle G$ have a maximum, and where a minimum? At what $\displaystyle x$ does the graph of $\displaystyle G$ have an inflection point?