Pode uma região possuir um limite de uma area finita?

Picture an unbounded region- i.e. one that extends in some direction forever. Because it just keeps on extending out, there's no way that it can have finite area, right?

WRONG! To show this counter-intuitive statement, let's look at the example: consider the unbounded region bounded by the curve $f(x)=\frac{1}{e^x}$, the $x$-axis, and the $y$-axis, which we'll call $R$.

Specifically, for any non-negative integer $n$, let's consider the section of the curve bounded by the points $(n,\frac{1}{e^n})$ and $(n+1,\frac{1}{e^{n+1}})$. Since $f(x)$ is always decreasing, this section of the curve is always on or under the line $g(x)=\frac{1}{e^n}$. So, the area between $f(x)$ and the $x$-axis in the interval $[n,n+1]$ is less than the area of the rectangle formed by $g(x)$, which we'll call $R_n$. This rectangle has length $1$ and width $\frac{1}{e^n}$, and so its area is equal to $1\cdot \frac{1}{e^n}=\frac{1}{e^n}$.

This bound for the area $R$ between $n$ and $n+1$ must hold for every possible value of $n$. So, we have that $$Area(R)<\sum_{n=0}^{\infty} Area(R_n)=\sum_{n=0}^{\infty} \frac{1}{e^n}.$$ This last sum is an infinite geometric series with ratio between $0$ and $1$, so we can use the geometric series formula to find that $$Area(R)<\sum_{n=0}^{\infty} \frac{1}{e^n}=\frac{\frac{1}{e^0}}{1-\frac{1}{e}}=\frac{1}{\frac{e-1}{e}}=\frac{e}{e-1}<\infty.$$ So, we just showed that the area of $R$ is finite (and bounded by $\frac{e}{e-1}$). In fact, there are many unbounded regions that have finite area, and we will explain when such a case happens in the next section.

Your friends from an infinitely far planet are heading back home soon. You want to be able to visit them one day, so you planned to build a cable car between their planet and Earth by attaching a string on their rocket as they take off. When they arrive at their planet, you will have your cable ready.

As you start to think about how to make this string, you realized that it has to be infinitely long! Wow, this is going to take a lot of material…

Finally, you got an idea. You took the paper, cut it in half along the longer side, and then cut one of the halves in half again.

Each time, you took one of the halves from last time, and then cut in half. Then, you attach the new paper strip to the end of the string.

You can keep doing it, while the new paper strips become super, super narrow – but you can always extend the string a little further. So long as you can cut the paper fast enough, and the string doesn’t break, you will be able to build a cable from Earth to that infinitely far place! This cable is one of the unbounded shapes with an infinite length and a finite area.

Isn’t it amazing – an infinitely long shape from just a single sheet of paper! This would be what we call an "unbounded" shape.

Try it for yourself. Take out a sheet of paper. How many times can you cut it in half along the longer side?