why isn't the surface area of revolution equal to the the sum of the perimeters of the circles that makes it? let $w$ be the width of a rectangle and $l$ be the length. From the definition of area it sounds logical to add up $w$ lengths of a rectangle to find how much two dimensional space it is occupying. When we carefully see what we did we take a one dimensional quantity and and added it some $w$ amount of times to get the area of this surface. Can't we do the same with three dimensional bodies. Doesn't it sound intuitive to slice up a revolution(a 3d object) into many 2d circles and find the sum of their perimeter to find the surface area of this 3d object.
*I am sorry I dont know how to properly put my idea in chronological order or use Latex to properly write a proof so I attached an image of my question. I would be happy if any one edits my question and writes it in proper form.
My question
The thing is I am wrong. To show that I am wrong we can do a proof by  contradiction. let $f(x)=x^2$.  By the already established method the surface area from $0$ to $a$ would be different from what my method would get which is infinity. I will put what I said in word in the next picture.
the illustration
So my question is Why am I wrong?, and why don't we use this intuitive idea?
 A: You can find the area of a solid of revolution by slicing it into circle and adding up the circumferences. But you have to be more judicious about how you slice.
The idea in the comments about wrapping wire rings around the surface is a good one.
The figure below shows two surfaces of revolution:
a cylinder, and a cone inside the cylinder.

All the wire rings are made from wire of the same thickness. But in the same distance along the axis of revolution in which we can fit five wire rings around the cylinder,
we can fit seven wire rings around the cone.
Conceptually, it's very simple. Take the curve from which you generate the surface of revolution. Mark off equal distances along that curve and put a circle at each mark. Now the sum of the circumferences times the distance between circles along the curve is a reasonable approximation of the surface area,
and becomes an accurate measure of the area once you turn it into a Riemann sum and take the limit as the distance goes to zero.
What you must not do is take circles whose centers are spaced at equal intervals along the axis of rotation and treat their circumferences as wires wrapped around the surface. The circles will be too far apart in the tapered sections of the surface.
Consider what would happen if we replaced the seven rings in the figure about with five rings around the same cone; there would be gaps between the rings, and we would not be covering the entire surface.
In practice, marking off equal distances along a curve is not always easy.
So instead we usually put the centers of the circles at equal distances along the axis of revolution, but multiply each circle's circumference by the distance between the circles along the curve, which is a larger distance in the tapering sections of the surface than in the cylindrical sections.
