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

 

[blue_raver22]

I can confirm that there is indeed a strong connection between integration and area. Integration is a mathematical concept that allows us to calculate the area under a curve in a given interval. This is known as definite integration and it is commonly used in mathematics, physics, and engineering to solve various problems involving area and volume.

To answer your question, yes, calculating an area in R^2 using integration would give the same answer as counting the unit squares. This is because the fundamental concept of integration is to divide a given area into smaller and smaller rectangles (or squares in this case) and then summing up their areas to get the total area. This process is essentially equivalent to counting the unit squares.

Now, you may wonder how integration was defined in such a way as to give the same result as most people's definition of area. Well, the concept of integration was developed by mathematicians over a long period of time, starting with the ancient Greeks and later refined by mathematicians such as Isaac Newton and Gottfried Leibniz. These mathematicians were able to develop a rigorous and precise definition of integration that is consistent with our intuitive understanding of area.

In essence, integration is a powerful tool that allows us to calculate areas, volumes, and other quantities in a precise and efficient manner. Its connection to area is a fundamental aspect of mathematics and has numerous applications in various fields of science. I hope this explanation has helped to clarify the connection between integration and area.