Positive and Negative Signs in Integration — Moosa Academy

Tell someone a room has an area of $-20$ square metres and it is nonsense. Tell them an integral came out as $-20$ and nobody blinks. Both statements are correct, and the reason is that the two are not measuring the same kind of thing.

Note Area as space, area as effect

Geometric area measures how much space a shape occupies. Space cannot run out below nothing, so that kind of area is never negative.

The area under a curve measures something else: the effect of the function on the system it describes. An effect has a direction as well as a size, and direction is exactly what a sign is for.

Note The water tank

Let

f

be the flow of water through a pipe into a tank.

Above the axis, water runs in and the level rises. At the crossing point the flow is zero. Below the axis the flow has reversed and water runs out.

Nothing about the lower patch is smaller. It is the same kind of quantity, doing the opposite job, so it enters the total with the opposite sign.

Theorem The sign convention

Where $f(x) > 0$ the region counts as positive area. Where $f(x) < 0$ the region of the same size counts as negative area.

f(x) \geq 0

[a, b]

, then

\int_{a}^{b} f(x)\,dx \geq 0

f(x) \leq 0

[a, b]

, then

\int_{a}^{b} f(x)\,dx \leq 0

The size of the lower region is

\left| \int_{a}^{b} f(x)\,dx \right|

The sign is decided by which side of the axis the curve sits on, never by how large the region is.

Theorem Two equal areas cancel

If a region above the axis and a region below it have the same size, they contribute nothing between them.

The tick marks say the two regions are congruent. One enters as

+A

, the other as

-A

$+A + (-A) = 0$

You filled the tank with a certain amount and then took exactly that amount out. The reading is zero because you finished where you started, not because nothing was ever there.

Example Reading a sign without calculating

A curve lies entirely below the axis on $[1, 4]$ , and the region it encloses with the axis has size $6$ . What is $\int_{1}^{4} f(x)\,dx$ ?

The curve is below the axis throughout, so

f(x) \leq 0

on the whole interval

Every contribution to the integral is therefore negative

The size of the region is

6

, so the integral carries that size with a minus sign

$\int_{1}^{4} f(x)\,dx = -6$

No antiderivative was needed. Knowing which side of the axis the curve sits on was enough to fix the sign.

Example Unequal regions

On $[0, 5]$ a curve encloses a region of size $8$ above the axis and a region of size $3$ below it. Find the integral, and find the total area.

The upper region contributes

+8

The lower region contributes

-3

$\int_{0}^{5} f(x)\,dx = 8 - 3 = 5$

For total area the signs are dropped and the sizes added

Total area $= 8 + 3 = 11$

In the tank picture: $8$ litres went in, $3$ came back out, the tank is $5$ litres fuller — but $11$ litres moved through the pipe.

Summary

Geometric area measures space, so it is never negative.
The area under a curve measures effect, and an effect has a direction.
Above the axis the effect is positive; below the axis it is the reverse.
The sign depends on the side of the axis, never on the size of the region.
Equal regions on opposite sides cancel to zero.
The integral gives the net effect; adding sizes without signs gives the total area.