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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.

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  • $\begingroup$ Because Riemann sums are actually over oriented intervals in $\Bbb R$. When you get to Lebesgue integrals you'll see what it means to integrate over an unoriented interval. $\endgroup$ – user137731 Nov 19 '15 at 2:38
  • $\begingroup$ So to say that curves above x-axis are +ve and others are -ve, is it oversimplification? $\endgroup$ – Seetha Rama Raju Sanapala Nov 19 '15 at 2:52
  • $\begingroup$ positive is +ve. negative is -ve. $\endgroup$ – Seetha Rama Raju Sanapala Nov 19 '15 at 2:58
  • $\begingroup$ Because the $\Delta x$ in the sum switches signs. $\endgroup$ – Ben Longo Nov 19 '15 at 3:03
  • $\begingroup$ @user3141822 No, "+ve" is not "positive", it's a weird thing that should never be written. $\endgroup$ – pjs36 Nov 19 '15 at 3:22
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This may help:

Why does an integral change signs when flipping the boundaries?

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.

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