Question: I need to find the volume of revolution of $$f(x)=\frac{2}{x+1},\;\; x\in [0,5],\;\;\text{about the x-axis}$$
In order to fully understand this question, one needs knowledge of understanding which shape to use to employ various methods used to solve these equations - such as the disk method and the washer method.
How do you understand which shape to use? Which method is used for this shape?
I started by graphing the function over [0,5], but I was lost from there.
Here, the disk method works just fine: a slice, or disk, with "radius" given by the function alone (since we're revolving around the x-axis which is given by $y = 0$).
$$\int_a^b \pi r^2 \,dx = \int_a^b \pi(f(x))^2\,dx$$
$$\int_0^5 \pi\left(\frac 2{x+1}\right)^2\,dx$$
The disk and washer methods are essentially the same --- the washer method, $\int_a^b \pi(f(x)^2 - g(x)^2)\,dx$, becomes the disk method if $g(x)$ is the zero function. The washer method is just a little more general. The disk/washer method is useful if you have a reasonable expression for the curve in terms of the axis parallel to the axis of rotation. For example, in the case you gave, $\frac{2}{x+1}$ (and its square) are easily integrable, so the disk method works great.
In other cases, it may be easier to use the shell method. For example, if $f(x) = y = \sqrt{x}$, and you are trying to revolve this curve about the $x$-axis, it is perhaps easier to rewrite it as $x=y^2$ and then use the shell method along the $y$-axis.
In many cases, either method works fine; it's a question of which is easier to apply. In other cases, only one method or the other can give an answer in terms of elementary functions.
