HSC Tips - Volumes (1 Viewer)

[McLake]

OK, tip page 7. I will try and make this more comprehensive than the last ...

Tips for Volumes:
- This topic obviously requires a strong knowledge of intergration, so make sure you know your integration stuff before you begin to study for it.
- The basic shapes include "bowls", "cylinders", "torus" (doughnut) but more advanced shapes (such as squares that become triangles or circles) are also inclued. It is imporatant that yopu knoe what the basic equation is for each type.


There are 4 basic ways to tacle questions:

- The easiest type is volumes of solids of revolution. This is appropriate for simple "bowl-like" shapes.
-- FORMULA: V = pi*I{a->b} (q - y)^2 dx or pi*I{c->d} (p - x)^2 dy where p/q is the offset.

- You must know how to do volumes by slicing, in both orentations (w.r.t. x and y), on the origin as well as offset. This is typically used for toruses.
-- FORMULA: V = pi*I{a->b} R^2 - r^2 dy

- You also need to know volumes by cylindrical shells, in both orentations (w.r.t. x and y), on the origin as well as offset. This is often used for spheres or "cylindrical" shapes.
-- FORMULA: V = 2*pi*I{a->b} x*f(x) dx [NB: This is like 2*pi*r*h]

- The last method, cross-section slicing, is used for irregular shapes. It requries you to determine a variable formula for the crossectional area, and intergrate this across the given region. Sometimes it will be neccsary for you to come up with your own axis to superimpose a shape onto, but this shouldn't be a problem. Simialr triangles are also often here.

 

[JizZ]

I was also told we needed to know the proof for cylindrical shells,

The volume (V) of a right cylindrical shell of height (h) and inner and outer radii (x and x + [delta]x) is given by:

V = pi*(x+[delta]x)^2*h - pi*(x)^2*h
= pi*h*{(x+[delta]x)^2 - x^2}
= pi*h*(2x*[delta]x+[delta]x^2)

If [delta]x -> 0, then [delta]x^2 may definitely be neglected.

V = 2*pi*x*h*[delta]x
= 2*pi*r*h*[delta]x

= 2*pi*I{a->b} x*f(x) dx

 

[Exeter]

^^ confirmation on that, we got told we NEED to derive it as well, not just simply state that V = pi*Ixy.d(x/y)

if all fails for a toros/square/hex/shape rotated around axis, simply find the area of the shape and multiply it by the central circumference, from the centre of the shape to the axis of rotation, might not get full marks, but better than nothing

 

[Affinity]

^^ Oh pappus's theorem the center of the shape means the centroid

SO.. WHAT THE... are we meant to derive the integral for spherical shells every question? *faint*

 

[Affinity]

By the way, can't you just prove the shell thing, by seeing the shell as a slab Dx thick and 2Pi*x * y in size folded into a cylinder? Fitz patrick and some other books do it that way.

 

[freaking_out]

Originally posted by Affinity

SO.. WHAT THE... are we meant to derive the integral for spherical shells every question? *faint*

well arnold...does this for every single question.

 

[ND]

Originally posted by Affinity
By the way, can't you just prove the shell thing, by seeing the shell as a slab Dx thick and 2Pi*x * y in size folded into a cylinder? Fitz patrick and some other books do it that way.

I did this in an assessment, and got no marks for it.

 

[Fosweb]

why cant you just do that?
because thats all a shell is...

 

[Affinity]

it is a little dodgy... but hey, it's in exemplar scripts and the MANSW booklet solutions..

 

[CM_Tutor]

The problem is that in opening the shell up and laying it flat there are minor distortions that take place. That is the volume of the shell is NOT equal to the volume of the rect prism but errors are of order deltax^2...thats why its a good idea if doing this to say
summing letting delta(x) go to zero AND ignoring terms in delta(x)^2

On this one, we are in complete agreement - all Extn 2 students should note this piece of advice.

 

[pitted]

ok i get where u r coming from and i understand it
however,
if we look at delta x being really small (which it is)
minor distortions cannot take place
y?
well think of it this way
in terms of quantum physics (i hope u understand)
now the line between quantum and classical is kinda fine but the errors in classical do not matter unless they are really big
e.g. equations involving the speed of light, measuring this or that (sorry i dont do physics)
what im trying to say iis
if the shell is in the form of a prism (and i hope that we talking about shells being turned into prisms then using that as the volume) there will be no distortions if delta x is sufficiently small
which may i remind u it is
for although there are inevitably distortions (im contradiciting myself i know) if they are small enough than summing for the volume would produce an error so small that it would not matter which way u used (thats why classical physics worked so well....sorry trying to fit that in and make sense of my argument)
thats why u end up with the EXACT same answer as doing it any other way
SINCE delta x is really small
i hope that was a clear argument (ok it wasnt sorry)
ciao

 

[aud]

Yea... I derive all the volumes... we were made to for every single volumes q ever, cause there could be marks in it... so like
Area = ...
delta v = Area with a delta x/y at the end
V = lim(delta x/y ->0), sigma, the delta v line
= integral line

Isn't that right? Learning more formulae is too hard... esp. if its a strange slice and you need to use the quadratic formula

 

[mojako]

they were talking about deriving the expression for cylindrical shell,
like in the pic.

I dont think we need to do that....

Although.. its easy once you understand it.

 

[Dumbarse]

use double integrals to solve volumes!!

 

[aud]

they were talking about deriving the expression for cylindrical shell,
like in the pic.

I dont think we need to do that....

Although.. its easy once you understand it.

Ohh... wait, but that's what I do... how else do you do it? Actually, don't answer that, it'll just make me more confused... I like the way I do it cause it makes sense and it's the only way I know how

 

[:: ck ::]

little shortcut (most of u know it but ill say anyway)

remember the area of circle formula when ur given the integral like sqrt(x^2-a^2) .. no need ot integrate unless they tell u to do so... but most cases they will just say "use the method of ..." nothin abt integrating =]

also simpsons rule comes in handy for parabolas etc...

 

[SmileyCam]

they were talking about deriving the expression for cylindrical shell,
like in the pic.

I dont think we need to do that....

Although.. its easy once you understand it.

oh dear god, I haven't derived a volume since we started its so much easier to just find the Area, then integrate it

 

[x.Exhaust.x]

- The basic shapes include "bowls", "cylinders", "torus" (doughnut) but more advanced shapes (such as squares that become triangles or circles) are also inclued. It is imporatant that yopu knoe what the basic equation is for each type.

What is the basic equation for each type? Any online sources? Looking through Patel gives me the creeps, as it's too difficult to understand.

 

[kwabon]

the only tip i have is to graph, and draw up your radius' and exerything, if you dont do that at least, it will quite hard to understand and work out the answer.

 

[gurmies]

A great tip for "irregular solids" would be using the fact that the relationship between the variables is always linear. This saves the hassle of drawing diagrams and utilising similar triangles (if they can be found).

PS: A very wise man shared this pearl of wisdom with me.