TT's Maths Thread

[brightsky]

I just found out both Chopin and Riemann died at the age of 39 and of the same cause of death: tuberculosis

fukn tuberculosis killed my fave composer and a great mathematician.

Riemann's your favourite mathematician? Hehe...Euler FTW!! But in the realms of music, Chopin wins hands down.

[TrueTears]

Nah he's not. I don't think I have a fave mathematician yet...

But Chopin <3 fave composer and most talented composer in my eyes.

[brightsky]

First remember the result

Now to find we need to find some polynomial containing .

To telescope: let

Now let

Thus

Now

But

Or you can directly use Faulhaber's formula for power sums.

[TrueTears]

mmm bernoulli would work well for sums of powers but telescoping would work better in general and requires more wishful thinking xD

zeitz style ftw

[brightsky]

Is there a way of using telescoping to find the general ?

[TrueTears]

http://vcenotes.com/forum/index.php/topic,19896.msg225557.html#msg225557

I'm haven't thought about that but I'm pretty sure a generalised version would involve telescoping at some point.

kamil has probably played around with it, ask him

[TrueTears]

1. The base of a solid S is a circular disk with radius r. Parallel cross-section perpendicular to the base are isosceles triangles with height h and unequal side in the base.

Set up an integral for the volume of S.

What the heck does the shape even look like?

2.

where do i place the axis.... too much art in this one...

3. A barrel with height h and maximum radius R is constructed by rotating about the x axis the parabola , where . Show that the radius of each of the barrel is .

Eh what is this question asking? " radius of each of the barrel" there only is 1 barrel? wdf

[/0]

My paint skills are no match for multivariable calculus... but imagine having a stack of sold, thin isosceles triangles lying around, each with the same height but different base lengths. If you place them standing up on a glass table so that one is in front of the other, and then look from underneath the table, you should see a circle. The circle is formed by their bases.

[TrueTears]

My paint skills are no match for multivariable calculus... but imagine having a stack of sold, thin isosceles triangles lying around, each with the same height but different base lengths. If you place them standing up on a glass table so that one is in front of the other, and then look from underneath the table, you should see a circle. The circle is formed by their bases.

Wait... but what do you do at the 'vertex' of the circle, you can't form an isosceles triangle...

I still have no idea what the shape looks like, all of this volumes and cylindrical shit is making me so confused.

Asif this is maths, it's like a fucking art class and i aint no piccaso

[/0]

My paint skills are no match for multivariable calculus... but imagine having a stack of sold, thin isosceles triangles lying around, each with the same height but different base lengths. If you place them standing up on a glass table so that one is in front of the other, and then look from underneath the table, you should see a circle. The circle is formed by their bases.

Wait... but what do you do at the 'vertex' of the circle, you can't form an isosceles triangle...

I still have no idea what the shape looks like, all of this volumes and cylindrical shit is making me so confused.

Asif this is maths, it's like a fucking art class and i aint no piccaso

lol picasso sucks at art

Anyway at the 'vertices' of the circle, you'd get a degenerate isosceles triangle (one with no area).


For 2) the easiest way is using a volume of revolution, but it also works in cylindrical coordinates, if you've reached that chapter yet.

For 3) the 'solid' is made up of an infinite number of barrels of different radiuses. The largest barrel has radius R.

fuck this im going to sleeep

[TrueTears]

wdf what kinda shape is that i cant even fucking visualise it, i keep thinking of a half hemisphere but the height must be the same WDF

maybe i need to get a degree in painting so i can paint some shapes to do these shitty questions


this is way too physics-esque, all of this cylindrical shell and volumes shit. where's the friggin maths

[moekamo]

2. (Image removed from quote.)

where do i place the axis.... too much art in this one...

for this one, flip the donut on the side and look side. This is the same as the pic of the circle attached rotated around the x axis

Make the circle equation: you'll also have to split it into top and bottom half of circles when you rearrange for

Does the volume end up as ?

[TrueTears]

Yeap it is, awesome explanation moekamo

I actually did it another way earlier today, used cylindrical shells, I thought that was easier to visualise than trying to set up an axis =S

[Ahmad]

What happens when R = 0?

[kamil9876]

http://vcenotes.com/forum/index.php/topic,19896.msg225557.html#msg225557

I'm haven't thought about that but I'm pretty sure a generalised version would involve telescoping at some point.

kamil has probably played around with it, ask him

Well i don't know about a quick general way, but it is an alright exercise to prove that it is always a polynomial of degree (n+1). And as an extra prove that the coefficient is (this allows u to find that integral).

By proving that, you have validated the method of just subbing in values into the polynomial and working out the linear equation system to find the coefficients.

Anyway a method I used to play around with this is:

let

This leads to the guess of a polynomial.


edit: oh yes, how could i forget. This can also be done combinatorily!