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.
Asked
Active
Viewed 105 times
1
-
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. – Nov 19 '15 at 02:38
-
So to say that curves above x-axis are +ve and others are -ve, is it oversimplification? – Seetha Rama Raju Sanapala Nov 19 '15 at 02:52
-
positive is +ve. negative is -ve. – Seetha Rama Raju Sanapala Nov 19 '15 at 02:58
-
Because the $\Delta x$ in the sum switches signs. – Ben Longo Nov 19 '15 at 03:03
-
@user3141822 No, "+ve" is not "positive", it's a weird thing that should never be written. – pjs36 Nov 19 '15 at 03:22
-
Why weird? It is most commonly used. Just shorthand for positive and negative. Nothing more or less than that. – Seetha Rama Raju Sanapala Nov 19 '15 at 03:49
1 Answers
0
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.
