I don't have a clear understanding of the relationship between area and the integral. I mean the nuts and bolts issues though I understand that the area under the curve of f(x) is given by its definite integral and that the areas above x-axis are taken +ve and those below -ve. My question is if we interchange the limits the answer changes the sign. The area remains above or below the x-axis. Why the answer is changing the sign>? Obviously, I am missing something important here. Kindly help. Also want to understand the principles clearly without much of maths. Plan is to once I understand the principles, I can get into the details of Lebesgue and other matters later. First I want to know the motivations and principles of positive and negative areas.
This may help:
The answers there are in greater detail than I could provide. It might be helpful to decouple the notions of integral and area; that is, the integral of a curve happens to be the area under the curve, but is not defined by it.