The connection between integration and area

[Poirot1]

My understanding is that (definite) integration defines area in R^2, volume in R^3 etc, and my question is: would calculating an area in R^2 using integration give the same answer as counting the unit squares. If so, how did they ensure to define integration in such a way as to give the same result as most people's definition?

 

[tkhunny]

If you are thinking "Unit Squares", you are not getting the idea of integration. They used to think this way in Ancient Greece - vaguely, if we can just quantize it, we'll find the fundamental unit. This is not good enough. The area of each "piece" really is zero!

We can get most peoples' definition to match up by teaching them THE definition. There are other definitions for integration - to handle discontinuities and various other things.

 

[CaptainBlack]

My understanding is that (definite) integration defines area in R^2, volume in R^3 etc, and my question is: would calculating an area in R^2 using integration give the same answer as counting the unit squares. If so, how did they ensure to define integration in such a way as to give the same result as most people's definition?

The Riemann integral is set up to be the limit if it exists of the area of sets rectangles of finite width that approximate the region under the curve as the widths tend to zero. So it is to a hand-waving approximation the result of counting squares as the size of the grid goes to zero.

It is silly to ask how did they ensure this result, the definition of the integral is set up explicity to match the intuitive idea of area in those cases where the intuitive notion makes sense.

CB

 

[Poirot1]

Thanks for the responses