Area, C of G of a paraboloid section? ...
Hi All, this is MathWimp here. I posted these queries in less appropriate thread categories here, before I figured out the website structure a bit more - mind you I have an excuse - the welcome E-Mail from the forum just invited me to post a homework question. I believe this to be a bit more involved than most homework problems:
I have an equation for a surface (ie: in 3D). Congratulations are unnecessary ;-) . Wonder if anyone can help with some problems related to it:
In an engineering application, there is a surface:
The surface is cut by 4 planes, all "parallel" to the z axis , 2 of them parallel to the y-z plane, 2 parallel to the x-z plane. The planes are , and , ; all 4 to be regarded as constants of the problem (which with makes 5 constants).
(ie: the projection of the cut surface on the x-y plane is a rectangle sides parallel to the x axis; and parallel to the y axis)
What is the area of the section of the surface cut off by these planes?
Note: I found out the hard way that the substitutions and will save lots of pencil-lead ;-).
I'm hoping for an analytic solution, but I can't find it, and I appreciate there may be no luck that one even exists.
If not, could you suggest a best or at least reasonable approach so that a "user" (say), would have a reasonably short paradigm so that given , and the planes , , , , the area of the cut-off section can be found?
And would it be possible to give error bounds on a non-analytical method? A TI 85 program, anybody?
In addition, though I've almost given up this, where is the C of G (ie: the x, y, z thereof) of the cut-off surface? (Assuming it composed of continuous material with uniform area density).
There are a couple of approximate approaches that have occurred to me, but I can't figure a way to tell how approximate they'd be (rather how accurate), so if it could be done more analytically, that'd be great.