Volumes in different dimensions

[r0bHadz]

Homework Statement

Does a line from f(x) to the x-axis have any area?

Relevant Equations

not sure how to describe the area of a line

I would assume that it has some area even if it is really really small. But I guess a line implies that the left and right boundaries are going to the middle an infinite amount, so it has area =0? does anyone get what I mean?

 

[FactChecker]

You are correct, but there is a way to represent that fact using $dx$ or $\delta x$ and $f(x)$. They might want that for an answer.

 

[r0bHadz]

You are correct, but there is a way to represent that fact using $dx$ or $\partial x$ and $f(x)$. They might want that for an answer.

oh wasn't a homework question, was just wondering myself. I appreciate it though! Also, I use to be able to mark my thread as solved with the old layout. Where is the button on this new one??

 

[SammyS]

oh wasn't a homework question, was just wondering myself. I appreciate it though! Also, I use to be able to mark my thread as solved with the old layout. Where is the button on this new one??

Looks as though there is NO such button now.

 

[fresh_42]

I would assume that it has some area even if it is really really small. But I guess a line implies that the left and right boundaries are going to the middle an infinite amount, so it has area =0? does anyone get what I mean?

No, the volume is zero. It has a length, but no area.

 

[WWGD]

To generalize, an n-dimensional object ( I think we need to assume completeness of measure) has , for k integer, k>0, (n+k)- dimensional measure equal to 0. The completeness is used to make sure it is measurable.

 

[Mark44]

No, the volume is zero. It has a length, but no area.

And to elaborate, a point is a mathematical object of dimension zero -- no length, width, or height.
A line is one-dimensional, and has length only.
A rectangle (or triangle, circular disk, or other similar plane object) is two-dimensional, and has area as an attribute.
A cube (or sphere or other solid object) is three-dimensional, and has volume as an attribute.

Things get more complicated if you allow fractional dimensions, with so-called space filling curves such as the Koch Snowflake or the Sierpenski Sponge. The Koch Snowflake can be shown to be a bounded curve (i.e., entirely contained within some rectangle in the plane) that has an infinite length. The Sierpenski Sponge can be also shown to have an infinite surface area, while being enclosed in some box in three-dimensional space.