How to explain that area under the curve is real number?

Question

As is known, one can consider area under the curve as area that consists of infinitesimally small areas bounded by $dx$ & $dy$ , in general terms.

Why doesn't there exist an area that equal to $r+\varepsilon$ where $r$ is real number and $\varepsilon$ is infinitesimal?

Eg : We can have $dA = dx \times dy$ & integrate it that within limits to get over-all area $A$. When $dA$ is itself infinitesimal , why should $A$ always be real number , without infinitesimals ?

How to explain that area under the curve is real number?

Downvoters, please explain where did I go wrong?

Thanks.

Answer 1

I am assuming that you are working in the modern framework of nonstandard analysis, then there is no problem: the rectangle with diagonally opposite corners $(0, 0)$ and $(r + \varepsilon, 1)$ has area $r + \varepsilon$. We can't expect area elements in nonstandard analysis to be standard reals.

Answer 2

At the risk of getting downvoted , I will give my thoughts on the matter.

(1) It is a Meta-level Philosophical Issue.
Consider the 2 Sets of Points.

What would be the "area" of the Green Part ? It will be "length times breadth" , though that is a Definition.

What would be the "area" of the Purple Part ? When we say it is "Same" , we are ignoring the "extra" line. When we say it is slightly more , we have to account for that "extra".

In one case , we are taking infinitesimals $\equiv$ ZERO.
In other case , we are taking infinitesimals $\equiv$ $\epsilon$.

Both Case , we have to ensure that the Definition is Consistent , not leading to Contradictions later.

(2) There is no "absolute" area , length & volume , except by Definition. We make the Definitions , which are consistent / interesting / useful / generic / logical.
When it suits us , we make signed areas which we can add & subtract.
When it suits us , we make areas always Positive which we can only add.
When it suits us , we calculate areas with vectors which have magnitude & direction , which gives us vector area !

We can , of course , make a new Definition where Points & lines have $\epsilon$ area.

(3) Normally , area is Defined by Axioms in Euclidean Geometry :

"Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists."

When we go with that thinking , Area must be a real number by Definition. There is no way to include infinitesimals at the macro level.

(4) Definite Integration generally goes with signed area when calculating area under real valued functions. We then have real area , with no infinitesimals.

[[ I have a bit more to add. Might take 12 hours more to update. ]]