Provide some rotations to obtain common shapes?

Permanence

I am attempting to use integration to determine the formula for the volume of three dimensional shapes for the sake of practice. My issue is I really lack the skills to determine what type of two-dimensional rotations will obtain a shape. The only one I have been able to do so far has been rotating a semi-circle over an axis to obtain a sphere.

Sorry if this is confusing, I'm not sure how to phrase myself. Better yet, I wasn't sure whether to post this in the calculus or geometry section.

Simon Bridge

The method of rotations will only be useful for things with a lot of cylindrical symmetry.
Finding the volume of, say, a dodecahedron, by this method won't work.

Permanence

Ah alright, thank you Simon. I'm just trying to keep myself entertained with this unit. The problems we're getting in class are so easily solved by an algorithm like approach I just don't even feel like giving them a look.

I was hoping I could keep myself busy by working on, what I consider, useful applications. I neglected a rather important issue though, so there goes that idea lol

Simon Bridge

Well, you could go figuring volumes in general - like: get a conical glass, (one that does not come to a point at the bottom) fill it with liquid, then tip the glass so the liquid pours out until the top of the liquid touched the opposite edge of the bottom surface to the lip of the glass.

What is the volume of the remaining liquid?

That should hold you :)

Permanence

Thanks again for the response Simon. I found myself starting at this problem for a good thirty minutes during an unrelated course, but made very little progress. I'm a bit confused about what is going and on the shape.

I imagined something similar to a wine glass, but I don't think that falls under the definition of conic. I also don't entirely understand your description of the event, could you try rephrasing the question or pointing me in the right direction?

With Regards,
Permanence

Simon Bridge

Yeh - the first thing you have to do is figure out what a "conical glass" is.
They can vary a lot - a straight-sided beer glass and a martini glass are both conical.
You could google "conical glass" and see what you get, then argue for a general geometry so that your equations will apply to any glass of the type.

Draw it side-on. The liquid-level makes a line from one side of the bottom to the opposite side of the top. If you like, try the same problem with a cylindrical glass first.

----------------

Aside:
A wine glass would be much more challenging - but if you held the wine glass horizontal, it would still have some wine in it. What's the maximum amount of wine such a glass could hold?
Consider: wine goblets can be spherical, elliptoid, or that special ISO wineglass shape.

This is very like the kind of problem you get in real life: you have to start by defining the terms of the problem before you can even start to work on it.

Permanence

Thanks again for the reply Simon. I'm swamped atm, but I'll give it a serious look over the weekend. The conical glass problem seems challenging, the wine glass problem seems darn near impossible lol.