volume of revolution
Posted 23 September 2005 - 01:41 PM
Simple question but I don't seem to find a formula for confirmation.
Suppose we have a curve y = f(x). The volume of revolution of this curve along the x axis is V = PI * Integral (y*y)dx = PI * Integral (f(x)*f(x))dx
If we use parametric coordinates, x = f(t), y = g(t), we end up with
V = PI * Integral( g(t) * g(t) * f ' (t))dt
Ok, all fine so far.
What about for a curve defined x = f(t), y = g(t), z = h(t). What is the volume of revolution of this curve along the x axis ? I cannot seem to find a formula. My guess is
V = PI * Integral ((g(t)*g(t) + h(t)*h(t))* f ' (t) dt
but I need this confirmed (book, web pagem anything)
Posted 23 September 2005 - 10:35 PM
Posted 24 September 2005 - 12:48 AM
Posted 25 September 2005 - 08:22 PM
Posted 26 September 2005 - 09:28 AM
In general the curve x = f(t), y = g(t), z = h(t) will not ensure the revolting it around an axis actually will generate something with a volume at all. All you know is that it generates a surface...
Posted 26 September 2005 - 09:40 AM
This would most likely be the correct approach. Because multivariable calculus can often is a pretty hairy subject for things like this, you probably want to get a 2 dimensional curve by taking your 3 dimensional one and doing something best described as 'looking at it from one side'; basically, if you want the rotation around the x axis, then ignore the z parameter entirely and perform the integration on the 2d curve that you're left with.
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