Why is surface area not simply $2 \pi \int_{a}^{b} (y) dx$ instead of $2 \pi \int_a^b (y \cdot \sqrt{1 + y'^2}) dx$?

We have an option to cut the solid of revolution (obtained by revolution of $y = f(x)$ between $x = a$ and $x = b$) into multiple slices in the following manner:

• each slice is a cylinder
• each slice is a section of cone cut by two parallel planes (a frustum of a cone)

Let the desired slicing be done via partition $$P = \{x_{0}, x_{1}, x_{2}, \dots, x_{n}\}$$ of $[a, b]$. We will apply both the approaches mentioned earlier to calculate the surface area as well as volume of the solid of revolution.

First we deal with volume which has an easier analysis. If we slice the solid as cylinders then the approximation of volume is given by $$V(P) = \pi\sum_{i = 1}^{n}\{f(x_{i})\}^{2}(x_{i} - x_{i - 1})\tag{1}$$ which is a Riemann sum for the integral $\pi\int_{a}^{b}\{f(x)\}^{2}\,dx$ and this is the desired volume.

If we slice the solid into frustums of cone we get the approximation of volume as $$V(P) = \frac{\pi}{3}\sum_{i = 1}^{n}\left[\{f(x_{i - 1})\}^{2} + f(x_{i - 1})f(x_{i}) + \{f(x_{i})\}^{2}\right](x_{i} - x_{i -1})\tag{2}$$ which is split into 3 terms and each term is a Riemann sum for $(\pi/3)\int_{a}^{b}\{f(x)\}^{2}\,dx$ so that the desired volume is again $\pi\int_{a}^{b}\{f(x)\}^{2}\,dx$

Let's now come to surface area of the solid of revolution. If we slice the solid into cylinders then the surface area is approximated by $$S(P) = 2\pi\sum_{i = 1}^{n}f(x_{i})(x_{i} - x_{i - 1})\tag{3}$$ which tends to $2\pi\int_{a}^{b}f(x)\, dx$.

If we slice the solid into frustums we get the approximation for surface area as $$S(P) = \pi\sum_{i = 1}^{n}\{f(x_{i - 1}) + f(x_{i})\}\sqrt{(x_{i} - x_{i - 1})^{2} + ((f(x_{i}) - f(x_{i - 1}))^{2}}\tag{4}$$ which can be simplified by the use of mean value theorem as $$S(P) = \pi\sum_{i = 1}^{n}\{f(x_{i - 1}) + f(x_{i})\}\sqrt{1 + \{f'(t_{i})\}^{2}}\cdot(x_{i} - x_{i - 1})\tag{5}$$ for some points $t_{i} \in (x_{i - 1}, x_{i})$. This can be split into two sums each of which is a Riemann sum for $\pi\int_{a}^{b}f(x)\sqrt{1 + \{f'(x)\}^{2}}\,dx$ so that the desired surface area is $2\pi\int_{a}^{b}f(x)\sqrt{1 + \{f'(x)\}^{2}}\,dx$.

We see that in case of volume both the approaches give the same answer. But in case of surface area the answers obtained by both the methods are different. Further note that out of the two answers we can easily verify which one is correct by using $y = x, a = 0, b = 1$ so that the solid of revolution is a circular cone. This verification shows that the technique used in equation $(4), (5)$ gives the correct surface area.

The question which OP is asking is this:

Why do both the approaches (using cylinders and frustums) give the same result for volume but different results for surface area?

The reason is simple. Both the sums in $(3)$ and $(4)$ are trying to approximate the surface area of the solid, but there is a huge difference between them namely $$\Delta = 2\pi\sum_{i = 1}^{n}f(x_{i})\left[\sqrt{1 + \{f'(t_{i})\}^{2}} - 1\right](x_{i} - x_{i - 1})$$ and this itself is a non-zero sum unless $f'(x)$ is identically zero. So the approximation $(4)$ is trying to take into account some additional surface area which is left out by sum $(3)$ and this additional part is significant unless $f'(x) = 0$ identically. Hence $(4)$ is a better and correct approximation.

In case of volume both the approximations $(1), (2)$ are Riemann sums for the same integral (but are expressed in slightly different ways).

In your link, it explains precisely why integrating $y$ doesn't work: Because the length $ds$ of a little piece of the arc is not $dx$, but $\sqrt{(dx)^2+(dy)^2}$, which can be written as $${\sqrt{(dx)^2+(dy)^2}\over dx}\cdot dx = \sqrt{1 + \left(dy\over dx\right)^2} \cdot dx.$$ It's the same reasoning why the length of a hypotenuse is $\sqrt{a^2+b^2}$ and not $a+b$.

Or this "proof" that $pi=4$: Is value of $\pi = 4$?

I cannot post a comment yet, but I will try to pick up where Carl Heckman left. The problem with the 'rectangle' approximation in this case, is the same as the idea of $\pi = 4$.

Consider the line $y=a x$, for example. Let $a$ become very large, and look at the surface area of the surface of revolution from $x=0$ to $x=1/a$. This should approach the area of a disc in the limit of $a \rightarrow \infty$, since we have a cone where the top angle will become very flat. However, if we were to put cylinders under this line and simply look at the surface area of the boundary of the cylinders, it is clear that we will not get the right answer, since this area will go to $0$ in this limit (the sum of the widths of the cylinders goes to zero).

We see that we somehow have to take into account the sides of the cylinders (perpendicular to the x-axis) as well. If we try to do that, however, we run into the problem Carl Heckman described. For example, look at the surface of revolution of a small part of the line around a point (not in the limit $a \rightarrow \infty$). $y$ does not vary much in a small enough area, so adding the sides of the cylinders as well, we would find that the surface area would be $$2\pi y(\Delta x+\Delta y) = 2\pi y|a+1| \Delta x$$

The error in this case is similar as to when you are trying to find the length of a hypotenuse of a right angle, by adding the two other lengths. Note also that $\sqrt{1+(y')^2} \Delta x$ is the length of the hypotenuse of a very small right triangle, which appears to be the factor we need to get the right surface area.

I hope I managed to make it slightly clearer why multiplying by $\sqrt{1+(y')^2}$ is the right thing to do without rambling on too much.