TITLE: Integrating left to right versus right to left.
QUESTION [6 upvotes]: OK, I understand that when integration is done left to right with respect to x increasing left to right (dx is positive), that the answer is positive, and vice versa when integrating right to left. But, as a reality check, there is something that I have never understood (or, understanding, I find so counterintuitive as to want to question it): Interpreting the result as area, one and the same area is either positive or negative not depending on what it is, but on how it is counted. I recognize that it is a logical extension of, say, -1 versus +1 on the number line, but at least -1 and +1 occupy different positions! Whenever I've brought this up, I tend to get blank looks, like, "how could you be so stupid", or "huh", but, clearly it does not keep me from being bothered by it, so I'll ask here ... once again! To me, apart from what the mechanics of integration require, a more logical "positioning" of the negative area would be its reflection across the x-axis.

REPLY [3 votes]: Although integrals can often be explained as "area", you are going wrong when you assume that integral is area. That is not true. Yes, they are related and numbers sometimes coincide. 
But take a look at sine integral - it's zero, although the circumscribed areas are clearly non-zero. Does that seem O.K. to you? You have to accept that integral is not area. It can behave differently and it can have different signs.
Have you had physics in school? If you are familiar with potential energies, this might help.
Take a look at such situation. Object with mass $m$ is at height $a$. It is lifted up to a higher height $b$. Force of gravity is $F=mg$. The object-Earth system has gained additional potential energy:
$$\Delta U_1 = \int\limits_a^b mg \mathrm{d}h = mg(b-a)$$
If the object was instead lowered from b to a then 
$$\Delta U_2 = \int\limits_b^a mg \mathrm{d}h = mg(a-b) = -\Delta U_1$$
Wither way the change in potential energy is the familiar $mg\Delta h$, but the change in height $\Delta h$ is opposite and so is the change in potential energy. Hope that's understandable and alright to you.
What I want to show, that integral is not the area function. It's more like accumulating operator. In the gravity case it accumulates energy by overcoming gravitational force while the argument $x$ (height) is changed. If you change it in other direction, the situation becomes obviously opposite.
Yes, if you take positive function and integrate it from left to right, you accumulate area and therefore you get the area. But if you go backwards, you are more like subtracting that area, therefore you get the minus sign.