The Integral of a Constant Function — Moosa Academy

This is the simplest integral there is, and it is worth doing slowly, because it shows what integration actually does. The rule $\int k\,dx = kx + C$ is not something to memorise — it falls out of stacking rectangles.

Definition The constant function

A constant function $f(x) = k$ has a horizontal line as its graph, crossing the $y$ -axis at $k$ .

f(x) = 4

is the lower line,

f(x) = 6

the upper one. The value of

x

makes no difference — the output never changes.

That is what makes the integral easy: the region under a horizontal line is made of rectangles.

Example Building the rule from rectangles

Take $f(x) = 4$ and walk along the axis one unit at a time, keeping a running total of the area collected so far.

x = 1

: one rectangle,

4 \times 1 = 4

x = 2

: total

8

x = 3

: total

12

x = 4

: total

16

Plot those running totals and they fall on a straight line. Its slope is $4$ — the very number we started with.

Repeat with

f(x) = 2

and the totals run

2, 4, 6, 8

— a line of slope

2

Repeat with $f(x) = 6$ and they run $6, 12, 18, 24$ — a line of slope $6$ .

Theorem The integral of a constant

$\int k\,dx = kx + C$

kx

is the equation of a straight line through the origin.

Its slope is

k

, the height of the constant function.

C

shifts that line up or down without changing its slope.

The rule is really a sentence about slopes: the integral of a constant is the line whose steepness is that constant. Every rectangle you add tips the running total by the same amount, which is exactly what a constant slope means.

Note What it looks like in practice

Water runs into a tank at a steady $5$ litres per minute. The flow function is the constant $5$ ; the integral function is the amount of water in the tank.

After

1

minute:

5

litres.

After

2

minutes:

10

litres.

After

t

minutes:

5t

litres.

A car holding a steady speed behaves the same way: speed is the constant function, distance travelled is its integral. In both cases the rate stays put while the total climbs in a straight line.

Example Using the rule

Find $\int 7\,dx$ , then evaluate $\int_{2}^{5} 3\,dx$ .

Here

k = 7

, so the integral is

7x

plus a constant

$\int 7\,dx = 7x + C$

For the second,

k = 3

, so

F(x) = 3x

F(5) = 15

and

F(2) = 6

$\int_{2}^{5} 3\,dx = 15 - 6 = 9$

The second answer is just a rectangle: height $3$ , width $5 - 2 = 3$ , area $9$ . The rule and the picture agree, as they must.

Summary

A constant function is a horizontal line crossing the $y$ -axis at its value.
The region beneath it is made of rectangles, so the area can be built by hand.
Accumulating those rectangles gives totals that lie on a straight line.
The slope of that line is $k$ itself, which is the whole content of the rule.
$\int k\,dx = kx + C$ .
A steady flow accumulates into height; a steady speed accumulates into distance.