Calculating the volume of a bottle of water using disk method

[Einstein44]

Homework Statement

I am trying to calculate the volume for a bottle of water using the disk method for integration. My plan was to take measurements and create a function for all the parts and calculate it like that. Now here are my problems:

1. Does anyone know any software that could help model the functions for the bottle at every part?
2. For the bottom part that is attached as a picture below, how can I calculate this? I have not been able to figure this out yet, so if anyone has an idea please let me know. For this I got told the following: "you can consider it separately and calculate the volume of one of the holes in the same manner as with the volume of the bottle, meaning you can model the hole's contour (you need to take measurements) and then use this function to find the volume using integration." However I did not understand what was meant by that.
3. For the middle part for example, I tried doing it by hand and considering it simply as a cosine function (if that is even correct), however: How can I find out what the vertical shift? Do I draw a line through the middle of the cosine function and measure how far it is above the x axis of the graph?
And then for the horizontal shift I don't know at all how I am supposed to know that, if there is any at all?
4. For the flat parts is it simply a constant function which you integrate??

Relevant Equations

the formula for the disk method is:

∫ π f(x)^2 dx

.

 

[256bits]

Worthy math exercise.
And then you fill the bottle up with water and do and check.

2. Get some play dough and press the bottom into it.
You might see the function better that way.
They mean take each dip, or hole, as a separate contour from the bottle.
Calculate the volume of a hole, and then multiply by number of holes to get the volume of the bottom for the bottle.

3. It looks to me that there are spherical contours along the bottle surface, such that you have
- bottom holes, flat, convex, concave, convex, a little bit of flat again, convex right up to the neck.

4. like a cylinder for the flat parts

An error in calculated volume would be the thickness of the bottle plastic. You could try with or without to get the exterior and interior of the bottle.

 

[Einstein44]

Worthy math exercise.
And then you fill the bottle up with water and do and check.

2. Get some play dough and press the bottom into it.
You might see the function better that way.
They mean take each dip, or hole, as a separate contour from the bottle.
Calculate the volume of a hole, and then multiply by number of holes to get the volume of the bottom for the bottle.

3. It looks to me that there are spherical contours along the bottle surface, such that you have
- bottom holes, flat, convex, concave, convex, a little bit of flat again, convex right up to the neck.

4. like a cylinder for the flat parts

An error in calculated volume would be the thickness of the bottle plastic. You could try with or without to get the exterior and interior of the bottle.

For point 3, how can I model these as a function then?

 

[256bits]

you do know the equation of a circle, right?
centered at the origin x² + y = r^r
and displaced from the origin? ...

For your problem,
you have to find the origin of the arc of the circle that represents the contour of the bottle, with limits x₁ and x₂, then find a f(x) for this curve.

So for a curved section you could have two volumes of revolution. There is a rotation of a volume for a cylinder if the arc designates a quadrant ( ie 90 degree arc ) - or, most likely, a trapezoid if the arc is less than that; plus the volume designated by revolution of the arc.
Your f(x) = volume of revolution of trapezoid + volume of revolution of the arc
Conceptually that is.

If you determine the equation for the arc, and do the integration for revolution of the solid you should find the volume.

And you have to do each section individually and add them all up for the volume of the bottle.

 

[Einstein44]

you do know the equation of a circle, right?
centered at the origin x² + y = r^r
and displaced from the origin? ...

For your problem,
you have to find the origin of the arc of the circle that represents the contour of the bottle, with limits x₁ and x₂, then find a f(x) for this curve.

So for a curved section you could have two volumes of revolution. There is a rotation of a volume for a cylinder if the arc designates a quadrant ( ie 90 degree arc ) - or, most likely, a trapezoid if the arc is less than that; plus the volume designated by revolution of the arc.
Your f(x) = volume of revolution of trapezoid + volume of revolution of the arc
Conceptually that is.

If you determine the equation for the arc, and do the integration for revolution of the solid you should find the volume.

And you have to do each section individually and add them all up for the volume of the bottle.

But to find the volume of the flat part (cylinder) I do not need to integrate anything, just using the formula V= hπr² right? Because that is already for a 3 dimensional shape ...
Although I am not exactly sure what you are referring to with the trapezoid. And what do you add the arc to that?
How are you representing the contour of the bottle using a circle (arc)?
I see you are trying to use geometry as opposed to functions in this case?

 

[256bits]

I see you are trying to use geometry as opposed to functions in this case?

Just for a visualization/

For the flat part the function f(x) = r, with limits x₁ to x₂
If you put that in your equation of revolution and integrate , you should get V_flat = πr²h, where h = x₂- x₁
Same thing.

 

[Leo Liu]

Homework Statement:: I am trying to calculate the volume for a bottle of water using the disk method for integration. My plan was to take measurements and create a function for all the parts and calculate it like that. Now here are my problems:

1. Does anyone know any software that could help model the functions for the bottle at every part?

Do you really need to calculate the volume for this specific water bottle? If not then you can well choose one whose volume can be easily calculated.

If it is difficult for you to measure the cavities on the surface of the bottle, maybe you can use a point light source to project the shadow of the water bottle on to a screen with grid, and then mathematically obtain the curve of best fit for the through matrices. While cool, this method might be imprecise though.

Your approach is brilliant -- but a little modification might make things slightly easier (maybe not): use a thin pen to trace out the circumference of the bottle on a piece of paper with 1cm grid. And then you can subtract the thickness of the pen in the inward normal direction (perpendicular to the tangent) to obtain the real circumference of the bottle. If you know Mathematica you may use its machine vision functions to find the curves. Yet there is no need to panic if you don't since you can pick the origin on the paper and then pick as many point on the sheet as you can, measure their position relative to the chosen origin, and then feed them into a spreadsheet to find the curves.

I hope you find this helpful.

https://math.stackexchange.com/questions/2305792/3d-projection-on-a-2d-plane-weak-maths-ressources

 

[Einstein44]

Just for a visualization/

For the flat part the function f(x) = r, with limits x₁ to x₂
If you put that in your equation of revolution and integrate , you should get V_flat = πr²h, where h = x₂- x₁
Same thing.

Yes, thank you. I got the right result. Using the formula for a cylinder I got that and using integration I integrated the wrong function at first, but now just using the radius I got the result. That part works. Now Ill have to find out a way to do the rest.
Thanks for the help!