Integração polinomial

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.

I'm only going to give you the basics since I know that you are clever and can figure out the rest. Now, that sounds scary, right? No, you really are smart enough to do it. Just remember that an odd function divided by an even function is odd and vice versa.

Is sin($x$) odd or even? Let's first check to see if it's even. Is sin($-x$) = sin($x$)? No, this can be easily verified to be wrong. Is sin($-x$) = -sin($x$)? Yes, this is true. So sin($x$) is an odd function.