Mean Value of a 3D Figure?

It means use the average value function.

Average height on the vertical axis is volume/area, which you can find by integration.

Reply 2

OP

Original post by NotNotBatman

It means use the average value function.

Average height on the vertical axis is volume/area, which you can find by integration.

I am having some confusion about the average height of solids through integration. Can you please refer me some website where I can read further about this topic?

Reply 3

OP

Original post by NotNotBatman

It means use the average value function.

Average height on the vertical axis is volume/area, which you can find by integration.

Are you sure it's not the same as centroid?

Reply 4

Original post by esrever

Are you sure it's not the same as centroid?

Yes, I think it means the same thing. If you have a function f(x) for the "elemental volume" (so that the volume =

\int f(x) \,dx

), and you then define

p(x) = f(x) / \int f(x) \,dx

to be a scaled version of f so that

\int p(x)\,dx = 1

, then the centroid (in x) is

\bar{x} = \int x p(x)\,dx

Compare with a probability density function p(x) in probability where we also have

\int p(x)\,dx = 1

and the average is given by

\bar{x} = \mathbb{E}[X] = \int x p(x)\,dx

and the similarities are striking.

Reply 5

OP

Original post by DFranklin

Yes, I think it means the same thing. If you have a function f(x) for the "elemental volume" (so that the volume =

\int f(x) \,dx

), and you then define

p(x) = f(x) / \int f(x) \,dx

to be a scaled version of f so that

\int p(x)\,dx = 1

, then the centroid (in x) is

\bar{x} = \int x p(x)\,dx

Compare with a probability density function p(x) in probability where we also have

\int p(x)\,dx = 1

and the average is given by

\bar{x} = \mathbb{E}[X] = \int x p(x)\,dx

and the similarities are striking.

I am a bit confused by the notation. Let's say y is rotated about the x axis from x = a to b. Is it correct to say that the mean value of volume is (volume)/(b-a)?

Reply 6

Original post by esrever

I am a bit confused by the notation. Let's say y is rotated about the x axis from x = a to b. Is it correct to say that the mean value of volume is (volume)/(b-a)?

No.

Edit: actually, I don't even know what you mean by "mean value of volume" - I think you're talking about something completely different.

(edited 4 years ago)

Reply 7

OP

Original post by DFranklin

No.

Edit: actually, I don't even know what you mean by "mean value of volume" - I think you're talking about something completely different.

I meant mean of a 3D figure formed by revolution

Reply 8

Original post by esrever

I meant mean of a 3D figure formed by revolution

As I said before, I believe mean would be synonymous with centroid here, although I concede the wording used in the specification does make one wonder if they mean something different.

I've googled and I can't find anything that both seems relevant and disagrees with it being the same as centroid.

@Gregorius @ghostwalker any other reasonable interpretations spring to mind?

Reply 9

@Notnek - any thoughts?

Reply 10

OP

Original post by DFranklin

As I said before, I believe mean would be synonymous with centroid here, although I concede the wording used in the specification does make one wonder if they mean something different.

I've googled and I can't find anything that both seems relevant and disagrees with it being the same as centroid.

For 2D figutes, mean is

\dfrac{1}{b - a} \int_{a}^{b} y \, \text{d}x

while centroid is

(\overline{x},\overline{y})= \left(\dfrac{\int xy dx}{\int y dx} , \dfrac{0.5\int y^2 dx}{\int y dx} \right)

Reply 11

Gregorius

14

Original post by DFranklin

@Gregorius @ghostwalker any other reasonable interpretations spring to mind?

Hm, bit obscure isn't it? A complete guess is that they're using "mean" to refer to the center of mass of a non-uniform shape or solid and "centroid" to refer to the geometric centroid (so the centre of mass of a uniform 2D or 3D shape). This is Cambridge International, right? They've removed it from the 2020 onwards syllabus, so maybe nobody understood what it meant :biggrin:

(edited 4 years ago)

Reply 12

OP

Original post by Gregorius

Hm, bit obscure isn't it? A complete guess is that they're using "mean" to refer to the center of mass of a non-uniform shape or solid and "centroid" to refer to the geometric centroid (so the centre of mass of a uniform 2D or 3D shape). This is Cambridge International, right? They've removed it from the 20120 onwards syllabus, so maybe nobody understood what it meant :biggrin:

Haha exactly. It's funny how they give us a formula booklet during exam where they include formula for sin(2x) = 2sin(x)cos(x) but don't include any of the centroid or mean formulas.

Reply 13

T!gger34

15

Original post by esrever

My syllabus states the following: Find mean values and centroids of two- and three- dimensional figures (where equations are expressed in cartesian coordinates, including the use of a parameter), using strips, discs or shells as appropriate.

I can find the centroid of a three dimensional figure but I'm confused about what is meant by 'mean' of a three dimensional figure.

What syllabus is this? Post a link if possible, please.

Reply 14

OP

https://www.cambridgeinternational.org/Images/329490-2019-syllabus.pdf

Original post by dextrous63

What syllabus is this? Post a link if possible, please.

Reply 15

Original post by dextrous63

What syllabus is this? Post a link if possible, please.

http://ciefurthermathematics.blogspot.com/2015/06/further-pure-topic-5-differentiation.html

(I remember searching this before and I didn't find something this helpful - mysteries of google...)

There still seems to be a lot of ambiguity; that pages states:

"The centroid of an object is such a point, that it is the average ( or arithmetic mean ) of all the points in the object."

and also

"the mean value of the function y = f(x) is given by:

\bar{y} = \dfrac{1}{b-a}\int_a^b f(x)\,dx

"

But it doesn't seem to ever actually define the "mean value ... of two or three dimensional figures".

My feeling (given the mention of strips, discs, etc.) is still that they mean "centroid" in the context of two/three dimensional figures. [Gregorius's caveat accepted, but at the same time, I think it incredibly unlikely you'd get a variable density object but be expected to find the centroid rather than c.of.m].

(edited 4 years ago)

Reply 16

ghostwalker

17

Original post by DFranklin

@ghostwalker any other reasonable interpretations spring to mind?

I would take mean, to be the average of some function, over a given area or volume. But just saying mean of a 3D figure doesn't make much sense without specifying what the mean refers to.

(edited 4 years ago)

Reply 17

OP

Thank you so much for help :smile:

. Can anyone also help me with finding centroid using shells? I have no idea what that means. Does it require me to use shell integration or is it a completely different idea?

Reply 18

ghostwalker

17

Original post by esrever

Thank you so much for help :smile:

. Can anyone also help me with finding centroid using shells? I have no idea what that means. Does it require me to use shell integration or is it a completely different idea?

I would think it would involve shell integration - never knew it had a specific name. Don't wish to try and teach it, but if you have a specific question you're stuck on.... Also, probably best in a new thread, as easier for everyone to follow.

Reply 19

OP