# Areas and volume ambiguity

For my question, I came up with this very simple analogy for the original question I have in mind.

Case 1 let's say there is a rectangle with the length $$X$$ and width $$Y$$, so the area will be $$XY$$. If I have $$2$$ of these rectangles The combined area would be $$2XY$$.

Case 2 Let's say I have a cuboid with length $$X$$, width $$Y$$ and height $$2$$, so the volume will be $$2XY$$.

Now my question is $$2XY$$ here as $$2$$ interpretations, so how am I supposed to distinguish that multiplying with $$2$$ gives me a $$2-$$dimensional answer or takes me into the 3rd dimension.

I came up with this problem when I was studying the integration by cylindrical shells method. That why do we get the thickness of the shell when we multiply it with delta $$x$$.

• Using units of length might help with a dimensional analysis Commented Sep 21, 2020 at 18:57

In the first case, the total area is $$XY$$. In the second case, the volume is $$2XY [L]$$, where $$[L]$$ is the unit of length you are using, e.g. meter, foot etc. This is assuming the $$X$$ and $$Y$$ include units along with the numerical value.

If $$X$$ and $$Y$$ are just numbers, the units are even more clear: $$2XY\ [L]^2$$, $$2XY\ [L]^3$$

In the first case the factor $$2$$ is a dimensionless number obtained by addition of two surfaces

$$XY+XY=2XY \quad [L^2]+[L^2]=[L^2]$$

while in the second case $$2$$ is a length times a surface, that is

$$2 \cdot X \cdot Y = 2XY \quad [L]\cdot[L^2]=[L^3]$$

Without specifying the dimensions involved is not possible establish a distinction between these expressions.

• does this not allow for higher than 3-dimensional objects? Commented Sep 21, 2020 at 19:04
• @blackmamba Yes we can extend to higher dimensions of course. When we are dealing with physical quantities indeed we need always to specify units.
– user
Commented Sep 21, 2020 at 19:08
• Okay then why does multiplying with delta x with 2pi.r.h gives us the thickness of the cylinder and not a 4-dimensional object. Commented Sep 21, 2020 at 19:12
• @blackmamba $2\pi r$ is the circumference, not area. Therefore, if you add along all $\Delta x$ (or integrate), then you get the volume of the 3d shape. Commented Sep 21, 2020 at 19:13
• In this case dimensions are $2\pi \cdot r\cdot h \cdot dx = [L]\cdot [L] \cdot [L]=[L^3]$.
– user
Commented Sep 21, 2020 at 19:14