Wrong Solution and Missing Picture

I'm actually still confused how r=x/2 comes about. Could we add a picture please?

Also, the solution tasks about cylinders. Really what we mean is 'discs', right? Cylinders sound like tall objects, discs are flat (dx). I think this is a more natural term and also what students would be more familiar. A 'cylinder method' may wrongly hint to thinking about shells (which may be an alternative solution, if we want to add that). At least it did confuse me.

We should be consistent with the terminology from the book/class, which is usually discs and (cylindrical) shells.

We actually mean a cylinder here - what you're doing is at some arbitrary point $x_{i}$ you're actually physically looking at a cylinder of height $\Delta x$ and of radius whatever comes from similar triangles. Really its not $dx$ but rather $\Delta x$ according to the textbook (and my lecture notes :D). Me not being from the applied school maybe these things are often interchanged but this is actually what we mean. Then of course you do the usual taking limits and then I suppose you end up with a disc I guess? But before taking this limit it actually is a cylinder - you are breaking up the cone into cylinder sections then finding the work done there then limiting. So I mean I'll change the solution to reflect this if we want - but I mean the general idea here is now correct and the solution as stated is correct (even though the textbook notation might differ slightly from the way it is/was stated in our wiki...)

I think everyone is speaking the same language here. We do interchange $\Delta {x}$ with dx in a limit because that's how the Riemann sum becomes an integral. So yeah the way you "picture it" is like stacking a bunch of thin circles on top of one another (or in your case cylinders with very small height) to build the cone (much like the favourite child toy with the little doughnuts that build the cone). I think we can all agree, that a lot of this language is funny to begin with and is designed so that we can do multiple integration in a single variable course. Perhaps the comment in regards to a more clear diagram etc. is just because often students get confused with height in an arbitrary sense (i.e. where to I put my axis) and height in a physical sense (i.e. you have to measure work from the ground due to the actual physical mechanism of gravitational potential energy).

sooooooooo.... is there any reason why we haven't changed this flag and added another one to the finished exam list?

I'm still not quite comfortable with the language. Cylinder seems to indicate something tall, I think disk would be better. Even if it's not completely flat, it does indicate something thin.

I'm with Erin. In Math 103 and Math 105 we would use "disk" instead of "cylinder". To make the point that the idea is to make them flat eventually. If you don't think cylinders is more natural for Math 101 students, then we will happily use the notation your class is used to. What I would like to see though before flagging it QG is a diagram making it cristal clear how r=x/2 comes about. In particular since there is more than one way of setting up the problem.

I hate to do this - because it will be the first and only time I ever hope to say a sentence like this but...

By Page 448 in Stewarts Calculus (7ed) second last line of text, they say "The ith layer is approximated by a circular cylinder with radius $r_{i}$ and height $\Delta x$"

This aside, I strongly disagree with calling this a disc. While it gives the right impression, its not right. A disc is a two dimensional object - we're measuring volume primarily not a 2-dimensional concept - calling this a disc is a bad abuse of mathematical language. This is a cylinder whose height we are thinking as limiting to 0.

Bernhard I've sent you a file to change the picture which maybe is clearer (though probably drawn worse). I'll leave it to you to upload (primarily because I don't know how yet).

Interesting. I have always been under the impression that a floppy/CD-R/DVD/Blue-ray disc was actually a physical object, hence three-dimensional. I don't think calling it a disc is a bad abuse. Apparently, even wikipedia and many book authors call it the 'disc method'. Googling for "cylinder method integration" gives you the method of cylindric shells, instead of the method used here. Hence the confusion. And if google says so, it must be right ;) Jokes aside. If you, as a math 101 instructor, say that you want to call it cylinders, then we call it cylinders.

I uploaded the picture and left both pictures in for now. I think the new image makes r=x/2 very clear, and hence I think the solution is QG now. Still I leave the review to Iain or Erin so that they can decide which picture they prefer. I don't think keeping both makes a lot of sense.

Carmen, our convention on how to add pictures to the wiki is here: http://wiki.ubc.ca/Science:MER/Adding_pictures

"floppy/CD-R/DVD/Blue-ray disc was actually a physical object, hence three-dimensional."

Its sentences like this that make me hate the English language lol.

So I guess its how you interpret it - when I think disc in terms of math I always think circle. (I guess though in life as you've pointed out things have different meanings...)

I mean really I don't know what to call this - these darn application questions always get me - they're just work problems - you break up the object into small parts and do the math on them - whatever they are - as this is the first time I've ever seen it and my entire experience with them is from this textbook - its all I know lol. I know you guys are probably much more seasoned than me at this (and thus I'm learning a lot from your point of views)... so I mean idk when I taught it I called it a cylinder because that's definitely what made sense to me. In any case we could argue about this for a long time but in the end I don't care - if someone wants to change the word cylinder to disc every time - that's cool. Either way - the answer is correct and that to me is the main point.

In any case thanks for the pointer Bernhard (Re: adding pictures)- clearly haven't had a lot of time to troll - I just add solutions to problems that don't need pictures :D

I made some small further changes such as moving the diagram to where I felt was a more appropriate place and removing the other diagram (if someone still needs it, here is the reference to it ). I then finally flagged it as good because I do think we've reach a sort of equilibrium in the conversation. I think the best approach now is to just let it be and see how students receive it, if there are complaints about cylinders/disks or origin at the apex/the bottom etc. then perhaps we will have further updates. At some point I may work on tidying up the diagram, but it's definitely usable in the interim. As a side note, a lot of the language I use with these types of problems is usually done in a way that is similar to how they look at them in physics classes as to avoid students thinking that they're two entirely separate topics. This is something that's hard to do for someone who has not had a lot of experience working with these application centered problems before.