Understanding Volume of Oblique Cylinders through Integration

[rishchat]

Hi :)

I'm doing my A-Levels and have a maths investigation project for which I decided to model the working of a shishi-odoshi. (http://en.wikipedia.org/wiki/Shishi-odoshi)

The shape of the water in the shishi-odoshi is a cylindrical segment and I want to use integration to find the formula relating its volume to the lengths of its sides and the radius of its base. I found the following page: http://mathworld.wolfram.com/CylindricalSegment.html which has everything I need but the working out is very brief and I can't follow what's going on.

I understand the final formula (1) by the intuitive reasoning given just above it but I can't understand either of the two derivations using integration. I know the bare bones of integration but I don't mind reading up to understand what they've done. The problem is I don't know what level of maths they're using or what topics I need to learn. If anyone who understands what they've done could perhaps explain it in a little more detail and tell me what I need to learn, it would be greatly appreciated.

Thanks :)

 

[Simon Bridge]

Welcome to PF;
Where did the description lose you?

They have considered the cylinder with it's base on the x-y plane (which is horizontal), and centered on the z axis (which is "up"). The segment is oriented so that the height of the cylinder varies with x but not y.

The analysis involves cutting the segment into thin slices either horizontally or vertically, finding the volume of each slice, and adding up all the volumes.

 

[rishchat]

Um, I've never heard of cutting a solid into thin slices to find it's volume; my knowledge of calculus is until Calculus I (I learned from: http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx) and the only way to calculate volumes that I've seen is the volumes of revolution method, which can't be applied to this solid.

So exactly what topic of calculus would that come under? I tried to google 'integrating over plane surfaces' (what it says on the page) but nothing came up :/

 

[Simon Bridge]

The method is usually taught after the level you have, but it is actually from the definition of "integration" through the limit of a Riemann sum.

http://www.phengkimving.com/calc_of...f_the_intgrl/12_03_finding_vol_by_slicing.htm

 

[rishchat]

I read through the link you posted and it was very helpful, thanks! :)

Now the only part I don't understand is the last step where they integrate the function (step 9 to 10) I've no idea how to deal with the squared x term under a root. All I can think of is to use the substitution rule but nothing I've tried eliminates all the x terms :/

What method could you use to integrate such a function?

 

[HallsofIvy]

I'm not clear what you mean by "the squared term under a root". Nor do I find any numbered "steps" so I don't know what you mean by "step 9 to 10". There were a number of example which involved square roots but using the "disk method" the function was squared eliminating the root.

It is possible to do integrals such as \int \sqrt{a^2- x^2}dx, \int\sqrt{x^2- a^2}dx, and \int\sqrt{x^2+ a^2}dx with trig or hyperbolic substitutions (x= sin(t), x= tan(t), x= cosh(t), etc.) making use of identities such as sin^2(t)+ cos^2(t)= 1 and cos^2(t)- sin^2(t)= 1. But I don't see any such cases in that link.

 

[Simon Bridge]

You mean the $V=\int_{-R}^{R}\sqrt{R^2-x^2}(ax+b)$ ... the a and b are constants to save typing.

Did you try substituting $x=R\sin(u)$?

 

[rishchat]

Haha, it seems like for every step of this project I have to learn something new...I guess that's kind of the point though :) So, I read up on integrating trig functions and trig substitutions and managed to integrate the function :D

Now, I need a function giving the location of the center of mass and they have formulae for the centroid (formulae 11-13) but I have no idea how they got them. They mention volume-weighted coordinates but a Google search yields nothing useful. How do you get the center of mass for a 3 dimensional object?

 

[Simon Bridge]

See: http://www.math.wpi.edu/Course_Materials/Calc5/Labs/node3.html
http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx
... plenty of others.
In your case, the mass density is a constant M/V

 

[rishchat]

While I'd love to learn about multiple integrals, there's simply too much prerequisite knowledge that I don't have. I used the page you linked to - Paul's Online Notes to teach myself integration and it has a section on center of mass (http://tutorial.math.lamar.edu/Classes/CalcII/CenterOfMass.aspx). It has formulae for center of mass in 2D that uses only single integrals, is there any such formula for 3D?

 

[Simon Bridge]

All volume integrals are triple integrals - taking slices is just putting dV=A(x)dx where you can get an easy function for A(x).

So for the x coordinate you could do $$x_{com}=\frac{1}{V}\int_{-R}^R xA(x)dx$$... Where R is the radius of the cylinder, M is the total mass, and V is the total volume already computed.
You'll see A(x)dx are the slices you took before. The trick is to be able to slice in all three directions.