Science talk:Math Exam Resources/Courses/MATH101/April 2010/Question 4

This solution is trying to access a file which it is having trouble doing. Also the solution should be 25pi/8, not 150pi/16. The error is that the solution measures height from the bottom (as most would) but the cone has its apex at the top so the similar triangle argument applies to the distance from the apex and therefore r=x/2 is wrong as it's actually r=(1-x)/2 so that the integral doesn't contain an $x^{3}$ but rather a $(1-x)^{2}x$.

It depends on how you view this but yes something is wrong - I think the easier fix would be to change the distance from $x$ to $1-x$ and make it $x^{2}(1-x)$ in the end integral.

The picture thing... That's a David or Bernhard thing :D

I should remark - This fix really just depends on where you put your origin. If you put it at the bottom or at the apex - the textbook for these students seems to arbitrarily use one or the other - I'm not sure which is a canonical choice - maybe eventually we can put both methods in...

Yeah, either method is fine, you either put your origin at the bottom in which case you have x(1-x)^2 or you measure from the top in which case you have x^2(1-x), both integrals are the same. I just wanted to point out that the answer was off by a factor of 3, no biggie.

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...)