Surface area (1 Viewer)

[largarithmic]

It's become fashionable it seems to write creative questions and post them here (and its a good and fun tradition too ^^) so here's one I wrote about a month ago, that generalises integration techniques for solids of revolution to surfaces of revolution. I don't actually know why this isnt taught in HSC because its cool and just as hard as volume.

Also since I wrote this quesiton Ive thought of another part, (from part d) is the sphere the only CLOSED/continuous (i.e. with zeros in the original function at the endpoints of the rotation) surface of revolution with this property?

Happy new year everyone

It's attached.

View attachment surfacearea.pdf

 

[Shadowdude]

Question: how did you write it up so pretty-ly?

 

[largarithmic]

Question: how did you write it up so pretty-ly?

I used LaTeX with a few addons (cant remember which ones exactly now)

 

[Shadowdude]

Umm... is that a computer program you had to buy or...? I've seen people use it, I'm never sure how they get it though.

 

[largarithmic]

Umm... is that a computer program you had to buy or...? I've seen people use it, I'm never sure how they get it though.

nah its freeware, http://www.latex-project.org/ftp.html has links to downloads. its pretty easy to learn too (most things are pretty intuitive) but for a lot of maths things you need to specific add-on packages at the beginning (but you dont need to download those, they're just extra packages of code already installed which dont need to be accessed for making a normal file)

 

[Carrotsticks]

It is disappointing that so few applications of integration are in the HSC.

There are volumes of revolutions and the occasional derivation of identities such as Stirling's formula.

But what about Surface Area of revolution? Arc length of a curve? Those topics are missing out on the fun.

 

[largarithmic]

It is disappointing that so few applications of integration are in the HSC.

There are volumes of revolutions and the occasional derivation of identities such as Stirling's formula.

But what about Surface Area of revolution? Arc length of a curve? Those topics are missing out on the fun.

You occcccaaaasssiooonnnaaalllyyyy do questions that are sorta like stirling's approximation (usually about harmonic series approximation) but I think theyre deemed too hard / too lateral

 

[Carrotsticks]

You occcccaaaasssiooonnnaaalllyyyy do questions that are sorta like stirling's approximation (usually about harmonic series approximation) but I think theyre deemed too hard / too lateral

That's the thing I don't like about the Syllabus. The subject is called 'Extension 2 Mathematics' for a reason, not 'Harder Extension 1'. And now they're dumbing it down further with the upcoming National Curriculum.

Integration is just an extension of integration, Volumes is an extension of volumes (derp), Conics is an extension of 'Geometry of the parabola' etc.

 

[largarithmic]

That's the thing I don't like about the Syllabus. The subject is called 'Extension 2 Mathematics' for a reason, not 'Harder Extension 1'. And now they're dumbing it down further with the upcoming National Curriculum.

Integration is just an extension of integration, Volumes is an extension of volumes (derp), Conics is an extension of 'Geometry of the parabola' etc.

Im yeah Id really prefer if the syllabus had more random stuff. like more geometry, more perms and combs, more inequalities, etc. although its not so much a syllabus thing, its more like, you can do cooler stuff with the syllabus. liek if the 8 question structure was changed so that each question focused on a particular idea and a lateral application of something in the syllabus, then thatd be a lot better. thing sorta is that like, with any syllabus there are gonna be "cool things" that you could do that are outside it; but there are also gonna be "cool things" that arent directly in it but the base knowledge to do them is there so they can be covered in an "investigative" style question.

 

[seanieg89]

Also since I wrote this quesiton Ive thought of another part, (from part d) is the sphere the only CLOSED/continuous (i.e. with zeros in the original function at the endpoints of the rotation) surface of revolution with this property?

I especially like this part. I proved that the only nontrivial functions f satisfying this property are semicircles IF we further impose the smoothness condition of f being twice differentiable (using differential equations). Am unsure as to whether or not there are any pathological non-smooth counterexamples to the statement.

EDIT: Just thought of one...We can glue together circular arcs. Will attach a figure later .

 

[tywebb]

Yes this is a good application of integration. Many of the surface area questions require substitutions, so it can't really be taught in 2 unit. Nevertheless Extension 1 students do substitutions and so it could be taught in Extension 1.

Here is some <a href="http://www.angelfire.com/planet/threeunit/arc-surface-notes.pdf">notes</a>, <a href="http://www.angelfire.com/planet/threeunit/arc-surface.pdf">exercises</a> and <a href="http://www.angelfire.com/planet/threeunit/arc-surface-sol.pdf">solutions</a> for practice.

 

[tywebb]

Another application is Gabriel's Horn and the Painter's Paradox



So a painter given the task of painting the horn using paint contained within it is faced with the paradox of only having a finite amount of paint to use but the impossibility of painting the infinite surface area with a uniform thickness of paint, however thin that may be!

 

[Trebla]

It is disappointing that so few applications of integration are in the HSC.

There are volumes of revolutions and the occasional derivation of identities such as Stirling's formula.

But what about Surface Area of revolution? Arc length of a curve? Those topics are missing out on the fun.

There's only so much content you can squeeze into one year...

 

[largarithmic]

There's only so much content you can squeeze into one year...

I reckon this shouldn't have been moved, the point of this question was take HSC maths and use it to do something random (in the way you use random complex numbers stuff to do cool stuff in this years q8)

 

[tywebb]

Yes. It could be done in Extension 1 after they have done integration by substitution.

 

[Carrotsticks]

It has actually been asked in a Trial before. I think it was Sydney Boys, I don't remember exactly. It was a 4 mark question.

"Show that the arc length of a curve can be expressed as *insert expression* "

I was extremely surprised when I saw that appear in the paper. I thought it was an incredibly cruel question, and in my opinion lazy on the examiner's part. The only people who would know what to do are those who have played around with Mathematics beyond the HSC.

 

[largarithmic]

It has actually been asked in a Trial before. I think it was Sydney Boys, I don't remember exactly. It was a 4 mark question.

"Show that the arc length of a curve can be expressed as *insert expression* "

I was extremely surprised when I saw that appear in the paper. I thought it was an incredibly cruel question, and in my opinion lazy on the examiner's part. The only people who would know what to do are those who have played around with Mathematics beyond the HSC.

Oh yeah I remember seeing that, 2003 or 2004 sydney boys test I think. I remember 2003/4 had ridiculous tests in general, the HSC was really hard in 03, and high, grammar, catholic trials were all pretty hard.

 

[tywebb]

I thought they were in the 2001 and 2002 papers. Here they are anyway:

sbhs2001 Q6b:

sbhs2002 Q8a:

 

[largarithmic]

I thought they were in the 2001 and 2002 papers. Here they are anyway:

sbhs2001 Q6b:

sbhs2002 Q8a:

oh okay. I didnt know there were two of them / my memory of this stuff isnt that great