How is it possible calculate Volumes from Adding together Areas?

Question

You can't find the volume for a $2D$ object, but by finding the area of say a square $A$ and multiply by any height $H$ you get the volume of cube with base $A$ and height $H$.

Here's my question. Is the volume of an object made of a very large amount of $2D$ surfaces stacked over each other? If yes, how is that possible? Since A $2D$ surface has thickness of $0$ isn't it like adding an infinite amount of zeroes and expecting to get a number that isn't zero?

Answer 1

You can in the same way you integrate to get an area. You can think of regular integration as adding up the area of small rectangles $y \times dx$. The line segment in $y$ has zero area, but you have a second dimension from the $dx$. Clearly this is an intuitive picture of what is going on. In three dimensions you can do the same thing-you integrate (area parallel to the $x-y$ plane$) \times dz$ to get a volume. The $dz$ thickens up the plane to make an infinitesimal volume.

Answer 2

This concerns calculus. Suppose you have a cylinder and want to find its volume. (Suppose also physics do not apply and everything is infinitely divisible). You could calculate the volume by multiplying the area of the base times the height. You could also divide the cylinder in a finite number of parts and repeat the process to find the individual volume of each part and add them together and the resulting volumes must agree. If you increase the number of cuts you'll decrease the height of each part while keeping the volume intact.

This procedure of increasing the number of cuts arbitrarily is known as a limit and specifically to calculate volumes and areas it is known as the integral. You would eventually get infinity times zero as you noticed but in the context of limits this is calculable with the fundamental theorem of calculus.

Answer 3

From what I understand, it is true that you cannot add something from one dimension together to create a new dimension, thus you are right in saying that adding a bunch of areas together will not give you a volume, it would only result in an extended area. What the others are hinting at is that calculus offers a work around by creating a completely new volume, albeit an infinitely small one, that happens to have the same base as the area you are working with. So in the end, you are adding up a bunch of volumes until it reaches a certain height.