Definite Integral as a Signed Area

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.