Definindo integrais a partir da area "conceitos"

We think of areas as positive, but when using integrals to calculate areas, they can be negative.

When an area is negative, it means it is below the $x$-axis.

Consider $y = 2x - 5$.

How can the area be $-4$?

Plot $y=2x-5$ from $0$ to $4$.

It starts out at $(0,-5)$, crosses over the axis at $(2.5,0)$ and ends at $(4,3)$.

The area of the section under the axis is a triangle, so it is $0.5\cdot 5 (2.5)= 6.25$ and the area above the axis is $0.5 \cdot 1.5 \cdot 3 = 2.25$.

If we consider the area under the $x$-axis to be $-6.25$ not $2.25$, the area turns into $-6.25+2.25=-4$, which is negative!

One last thing- when areas are positive/negative, we call them signed. This is because they can possess positive/negative signs.