Proof to me the area of a square is l*w

Proof to me the area of a square is l*w

It's defined to be that so there's nothing to prove.

A RECTANGLE can be represented by a constant height and a set width, then we can define the area as
[math]A=\int_0^l h dx=hx|_0^l=hl-h\cdot 0=hl[/math]

Length and width are both one dimension each. Area is two dimensional with both length and width.

Not a proof. Since with integrals each Infinitesimally small rectangle(dx*f(x)) is defined to be length * width.

No, this is the Lebesque measure.

In fact, it would be neater to define it as
[math]\int_AdA=\int_0^l\int_0^hdxdy=\int_0^ldy\int_0^hdx=lh[/math]

Lebesgue measure on R^2 is the inf of boxes containing, with the measure of a box being lw
So this is also circular

Well, the R^2 Lebesgue measure is defined as the product measure of two R Lebesgue measures, where you assume that the measure of (a,b) is b-a. Nothing about boxes.

Well it's easy user you can do it yourself

>draw a rectangle
>measure length and width
>now measure the area inside
>if it's equal to lxw draw another rectangle
>repeat above for all possible rectangles
>if you don't find an exception than lxw is the area of a rectangle
>if you do you'll be famous

Baseless assumption like in most of modern mathematics

Integral of y=a in the range x=[0,a]

one gallon of paint (G) can cover an area of 400 square feet (1)
let's say we have a wall with l = 20 ft and w = 20 ft
by contradiction, we try to prove that l*w is not equal to the area of the square, then either G will come up short or there will be some paint left over
we paint the wall, and observe that all paint will be used, and the wall will be completely covered
then we know by (1) that the area is 400 square feet, in other words l * w

I call this an engineer's proof

that won't get you far in the math olympiad I'd say

You dorealize that the dedfinition of the integral uses the area of rectangles right?

Not even the same guy, but he said that x = y = a.

meanwhile, mathfags are still arguing over the definition of the integral

just draw a rectangle made up by many squares and then count all of the squares

The thing is, you can't. You have to start from the axiom that the area of a 1 x 1 square is 1. Without that assumption you can't prove it

Area is space contained by shape
Shape is determined by x sides, at y length and z width.
Sqaure is 4 sides all equal
If borders determined by length and width
The area within sqaure should be the measure of length in units widths

I'm an engineer and this offends me.
First of all, real engineers use metric units. I'll ignore this for the sake of demonstration.
Secondly, an engineer doesn't dirty his hands by actually painting the house. He also doesn't trust painters to do their jobs correctly so that's out of the question.

A true engineer will use rounding to his advantage! 400 square feet are around 1e3 square feet. The dimensions of the wall would be rounded to 1e1 x 1e1 feet.
As we can clearly see, 1e1 x 1e1 = 1e2 square feet of wall.
1e3 sq ft >> 1e2 sq ft which means:
>go to boss
>tell boss that because A = L x W we would be able to JUST ABOUT cover the wall in paint (20 x 20 = 400)
>go into storage
>get a container
>pour most of the paint into container, leaving only enough for 1e2 square feet since we determined that to be the ACTUAL area of the wall (see above) + 20% margin of error
>steal our container of paint
>suck the painter's dick so he'll use the stolen paint to paint my own house

Take 6 sticks of equal length. Lay three parallel and touching. Make sure their ends are lined up. Now lay the other three on top, but perpendicular in alignment to the original three. Also parallel to each other and the ends all lined up.

Is it a square? Are all four of its sides of equal length? Now, count the sticks.

Q.E.D.

So there's not a drop of paint left in the can and the wall is uniform in density of coating?

Get out.

You need to define area first, otherwise it's impossible.

Think of how many 1x1 squares you can fit in the square of interest. You can fit l across the L axis and w up the W axis.

Just keep shrinking the peppers. Different pepper sizes mean different units. They still all fit in the same area. Now imagine infinite infinitely small peppers. Call one row of those peppers length and call one column height then multiply them.

Good, but that only works when L and W are integers

Why do I have that many peppers