Is the area of a line = 1? I tried to teach my son multiplication using a rectangle.
(e.g. 3cm * 4cm = 12cm^2).  Now I have 12 little squares.  But how do I explain where the "little square" came from?

My best guess: If I cut a little square in two, I can get a rectangle 2cm long * 1/2cm deep.  Then I cut that rectangle in two, I get 4cm long * 1/4 cm deep.  I keep going and say I get a rectangle 100,000cm long by 1/100,000 cm deep.

This is very long and very thin.  And I deduce, that the limit would be a line.  And so I conclude that the area of a line = 1.  (units become irrelevant, infinity takes care of that).

I also like this to explain why the integral of y=0x+1 (for x from 0 to 1) is 1.
Note: this should be a 1 unit square, and integrals are an area but we say the answer is 1 not 1 unit^2.

My question: Do you agree? (if not, why not?)
The ‘little squares’ are simply the result of dividing each side into $1$ cm segments and using those to form a grid of $1\times 1$ squares within the $3\times 4$ rectangle. There’s no reason to look any further for an explanation of where they come from. We define the area of a $1\times 1$ square to be $1$ square unit (where unit is whatever linear unit was used to measure the sides of the square), so the obvious way to find the area of some other shape is to divide it up into $1\times 1$ squares $-$ if we can.

Most of the rest is simply wrong, I’m afraid. For starters, a straight line has no area. If you want to explain geometrically/pictorially why a rectangle of dimensions $\frac12\times 2$ has an area of $1$, divide it into $4$ squares of dimensions $\frac12\times\frac12$ and observe that they can be rearranged into a single $1\times 1$ square, which by definition has area $1$ square unit. The same reasoning applies to a rectangle of dimensions $\frac1{100}\times 100$: since $100=10000\cdot\frac1{100}$, it can be divided into $10000$ squares of dimensions $\frac1{100}\times\frac1{100}$, which can then be arranged in $100$ rows of $100$ squares to form a unit square. There is no limiting procedure involved here.

Finally, $\int_0^1 1~dx$ is $1$ square linear unit (e.g., sq. cm.) when it’s interpreted as the area between the $x$-axis and the line $y=1$ over the interval $0\le x\le 1$; it has other units when the integral is given other interpretations.
Lines simply have no area. I'm still a bit confused by the fact that you managed to conclude that the area of a line is 1, and I don't know why you mentioned infinity. Use basic logic. A line has only one spatial dimension, and a square has two. Let's say that the area of that square is 4. If so, then we know that each side must measure at 2. Since we know that to find the area of a square you must 'square' on of the side's measurements, we form the equation l x w = a. A line, by definition, can only have either a length or a width (whichever you prefer to call it). The other dimension has a value of 0, or none. So, no matter how long a line is, it will never have an area because one measurement is missing. Having 'no' measurement is different than just having '0', however. A line's missing measurement can not be 0, and here's why. If the length of a line is equal to infinity (meaning that it is a true line and not just a segment), then that length of infinity must be multiplied by 0. Since 0 multiplied by infinity is just infinity, then the area of the line would be infinity. We know this is not true, which is how we conclude that a line has no second dimension.