Why does an integral change signs when flipping the boundaries? Let us define a very simple integral:


*

*$f(x) = \int_{a}^{b}{x}$ 


for $a,b\ge 0$.
Why do we have the identity $\int_{a}^{b}{x} = -\int_{b}^{a}{x}$?
I drew the graphs and thought about it but to me integration, at least in two-dimensions, is just taking the area underneath a curve so why does it matter which direction you take the sum?

There is nothing to prove here. It is just a definition (more precisely, just a notaion). Note that for $b \geq a$, from the point of view of Lebesgue integration, the value of the integral only depends on the domain $[a,b]$ (and does not depend on the fact that if we consider the function from $a$ to $b$ or $b$ to $a$) and this value is written as $\int _a^b f$.
The point is that if we make such a definition (or notation), then we can write the calculations involving the change of variable formula for integration in 1D more conveniently. More precisely, if we have such a definition, then we do not need to consider separately the two cases, when the change of variable function is monotonically increasing or decreasing.
