The Integral of a Constant Function

Where ∫k dx = kx + C comes from. The region under a horizontal line is made of rectangles, so the rule can be rebuilt by hand: the running totals lie on a straight line whose slope is k itself.

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This is the simplest integral there is, and it is worth doing slowly, because it shows what integration actually does. The rule is not something to memorise — it falls out of stacking rectangles.

Definition The constant function

A constant function has a horizontal line as its graph, crossing the -axis at .

6 4 x
is the lower line, the upper one. The value of 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 and walk along the axis one unit at a time, keeping a running total of the area collected so far.

4 1 2 3 4 x
At : one rectangle,
At : total
At : total
At : total

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

4 8 12 16 1 2 3 4 x
Repeat with and the totals run — a line of slope .

Repeat with and they run — a line of slope .

Theorem The integral of a constant

is the equation of a straight line through the origin.
Its slope is , the height of the constant function.
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 litres per minute. The flow function is the constant ; the integral function is the amount of water in the tank.

After minute: litres.
After minutes: litres.
After minutes: 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 , then evaluate .

Here , so the integral is plus a constant
For the second, , so
and

The second answer is just a rectangle: height , width , area . The rule and the picture agree, as they must.

Summary
  1. A constant function is a horizontal line crossing the -axis at its value.
  2. The region beneath it is made of rectangles, so the area can be built by hand.
  3. Accumulating those rectangles gives totals that lie on a straight line.
  4. The slope of that line is itself, which is the whole content of the rule.
  5. .
  6. A steady flow accumulates into height; a steady speed accumulates into distance.
283. Relations and Functions
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