Using a formula to prove cyclindrical shells (1 Viewer)

[ToO LaZy ^*]

Using a formula to prove cylindrical shells

i was wondering if you could just prove the formula for the cyclindrical shells and then say..
"hence, using this forumla V = 4piS...."

here's the actual question..can someone show me the proper way to do these if using the formula isn't sufficient enough
tank u

 

[ngai]

i was wondering if you could just prove the formula for the cyclindrical shells and then say..
"hence, using this forumla V = 4piS...."

lol y not?
it does say 'using cylindrical shells'
so i guess ur meant to use cylindrical shells?

 

[cj_bridle]

yeh i think the point is just to find volume of shell then sum it...

u dont need to prove the volume of the shell.. just use the formula..

i dont see why you couldnt do it your way.. but the normal way seems more logical imo

 

[ToO LaZy ^*]

yeah, it says:
'show using the method of cylindrical shells, that the volume V...'

could you show me the proper way if it's not sufficient..(we've only had 2 lessons on the volumes topics and it's in the trials comin up on monday )

 

[CrashOveride]

We have the radius as x
The height as 2sqrt(4 - (x-3)²)
Now A(x) = 4pi.x.sqrt(4 - (x-3)²)

V = lim deltax --> inf Sigma (from 1 to 5) [Area(x)]
Thus volume V = 4pi INTEGRAL 1-->5 x.sqrt(4 - (x-3)² dx

Are you having trouble with the second part ?

 

[ToO LaZy ^*]

well both parts..mostly part b tho

 

[CrashOveride]

You could use the substitution, if you'd like i could post that up. However, because we can prove it via the "otherwise" option given, I would turn to Pappus' theorem which gives me the volume as 24pi² units³

 

[ToO LaZy ^*]

yes please!!..(i'm on msn if it makes things a bit easier..)

and what is this pappus theorem you speak of?

 

[CrashOveride]

Using the substitution, your new limits are from -2 --> 2. dx=du etc. substitue in for values of u etc. You will notice of the the two integrals is actually an odd function, going from -2 --> 2 which means it is zero. The other one can be found quickly by remembering that it is a representation of a semi-circle.

In any case, V = 24pi² cubic units.

 

[cj_bridle]

without recognising that the 2nd one is a semi-circle what would the integration of sqroot(4-u^2)du ... be? the only way is through substitution again isnt it?

 

[ToO LaZy ^*]

i used the substi. but it got very very messy..unless i did something wrong...

 

[cj_bridle]

mine got messy aswell..

nop dont worry mine didnt get messy i just forgot to change limits no wonder i got a -ve sqroot

 

[ToO LaZy ^*]

mine got messy aswell..

nop dont worry mine didnt get messy i just forgot to change limits no wonder i got a -ve sqroot

but you still had to use IBP right?

 

[CrashOveride]

No you shouldn't need to use IBP in this scenario.

 

[mojako]

where did you know that it's called the Pappus' theorem ? your teacher or a textbook (and what textbook just for my information)?

you need to show that you understand where the formula comes from.
draw a rectangle but with a thickness of delta(x).
write its height and its width

A more formal proof is illustrated in the attachment, for a different question.
you can do the general case, I'm just too lazy to do that.
In this picture, radii are "R" and "R+delta x"
You can also choose the radii as "R-delta x" and "R", or
"R+0.5delta x" and "R-0.5delta x", or something else.
you'll end up with the same result.

But, you don't need to show this proof in your answer to the question you gave, as long as you draw a picture.
The large amount of marks (8 marks) is likely to be given because it might be harder to recognize the easy way to integrate that, because if you don't you'll have to spend a long time doing that Q.

 

[CrashOveride]

I never rely on any formulae when doing volumes. I think to play it safe you should learn how to do all questions from first principles, it's not so hard either.

mojako: I learnt of Pappus' theorem through a really old volumes book my teacher gave me. My standard four unit book, along with others I believe, don't mention this. It's quite a handy tool to have at one's disposal, but as it's not mentioned in the syllabus, one should use it only for "checking" (or if one is desperately short of time).

 

[mojako]

Bill Pender's Cambridge 3U mentions that technique in the integration by substitution section, but doesn't mention the name (i.e. Pappus' theorem).
To confirm, what you meant by Pappus' theorem is simply changing the boundaries of the defnite integral right?
I'm quite sure it can be used in exams, not just for checking. That's how people do integration by substitutions involving definite integrals at my school.

What exactly is integration by first principles? Just putting the lim and sigma notations?

 

[shazzam]

hey it's probably in coroneous since
1. those books are pretty old
2. Pappus and Coroneous both sound like Greek names so maybe Coroneous learnt it when he was in school *shrugs*

There's my two cents