Calculus - Cylindrical Shells Method

[MG-Cloud]

Hi,

I have a test tomorrow and am having a huge amount of difficulty understanding the cylindrical shells method of calculating volume.

Basically (from the ntoes I have),

V = 2pi * integral from a to b of (x * f(x))dx.

I understand this formula enough to plug things in and solve - however, sometimes I plug in the wrong things.

The only non-constants in here seem to be x, and f(x). I always just plug in x as x, and the given function directly into f(x). This seems to work 90% of the time. However, sometimes it doesn't.

Therefore, my questions are -

What would the circumstances be in which x or f(x) are not just x and the given function?

Finally - the reason I'm having so much trouble with this is that I don't really understand what's going on - I can't visualize it like I can for the other volume methods. Any insight on this would be *greatly* appreciated.

Thanks!

[clonk]

A more accurate formala would be:

Intergral From A to B of 2 * pi * (shell radius) * (shell height) dx

When you revolve the shape around the y-axis the shell radius is the value of x, but if the shape is revolved around the x axis you will need to express your figures in terms of y.

I find the shell method the hardest to visualize as well - my book describes why as follows: There is another way to find volumes of solids of rotation that can be useful when the axis of revolution is perpindicular to the axis containing the natural interval of integration. Instead of summing volumes of thin slices, we sum volumes of cylindrical shells that grow outward from the axis of revolution like tree rings.

Thats about all the help I can give you, shells were the toughest of the methods for me too.

[MG-Cloud]

Wow great! That's a lot more info than in my notes =) Thanks so much - I'm gonna go work on this right now =)

[lostspyder]

I think your problem is that you feel confined to the equation. All the equation is there for is to help you figgure out how to do the meathod. If theres one thing about high level math that you have to learn is that you dont get to plug and chug anymore. The eqations get simpler as the logic becomes harder.

Picture the function as a set of rings with a very small hight. These rings might have a hole or not (dependsing on the function). Now find the volume of one of thoes rings using geometry. with the radius in terms of a function (possibly modified due to a shift (ie if you take the axis of revelution around 1 instead of 0)) and dx or dy depending on what axis is the hight. Now you simply intergrate over the hight.

Once you start looking at it abstractly youll have no problem, i garuntee it.

Not shure how much help this may be, im not very good at explaing things.

[MG-Cloud]

I think your problem is that you feel confined to the equation. All the equation is there for is to help you figgure out how to do the meathod. If theres one thing about high level math that you have to learn is that you dont get to plug and chug anymore. The eqations get simpler as the logic becomes harder.

Yeah, I realize that :) The problem was though I missed the class, so I only have notes which aren't very complete - as long as I have an idea of the concepts, I can generally derive many things from them.

Once you start looking at it abstractly youll have no problem, i garuntee it.

Not shure how much help this may be, im not very good at explaing things.

Your post was a great help - thanks! :D