How can I get the full formula to describe a solid of revolution?

Let's say I got the function $f$, and $y = f(x)$, how can I get of formula that give me $z$ to get the function that create a solid of revolution around any axis(the $y$ axis ? the $x$ axis? an arbitrary $x$ or $y$, a $mx+p$ type function... Any straight line in the space ? ) and on a interval $[I,J]$?

Does it exist, and if yes, how, a method to, just by that, get $z$ to describe the function in the space (something like $y= f(x); z= ...$).


Well, it makes sense to use integration to find the formula for the solid/volume of revolution. Let's say you've got the function $y = f(x)$ as you say. We can imagine that at any one point on the x axis, we can draw a circle by drawing a line straight up the y axis and rotating it round 360 degrees. This circle's radius would be $\pi$ so the area of a circle drawn at any point on the line $ f(x) $ is:

$$ Area = \pi y^2 $$ but since... $ y = f(x) $ $$ Area = \pi f(x)^2 $$

We can also say that the summation of all the single, one-dimensional lines drawn up the y axis at ALL the points of x in the interval $[I,J]$ using calculus integration is:

$$ \int_I^J f(x)\,dx $$

Now we have our area, we just rotate it around the x axis through 360 degrees:

$$ Solid/Volume\,of\,Revolution = \int_I^J\pi f(x)^2\,dx $$

But since $\pi$ is simply a constant:

$$ Solid/Volume\,of\,Revolution = \pi\int_I^J f(x)^2\,dx $$

  • $\begingroup$ That give me the volume, that's a good point, but I guess this isn't the $z$-formula who give the solid in any grapher. And if I understand good, that is only for a rotation around the $x$ axis, what a $y$ axis rotation would be? $\endgroup$ – Robin Carlier Oct 27 '14 at 18:31
  • $\begingroup$ Hey, think about it - integrating the inverse function gives you the area relative to the y axis, so you need to use the same formula except with $f(x)^-1$. So... $$ Solid/Volume\,of\,Revolution = \pi\int_I^J f(x)^-2\,dx $$ $\endgroup$ – Resquiens Oct 27 '14 at 18:36
  • $\begingroup$ By the way, this would definitely always give you the volume - I believe it's as general as you can get. As for a volume relatively around any other function - I'm not sure on that point, but there must be a more general form which in turn gives the formulae for rotations around the x and y axes as well as any other function (for example a straight line). Perhaps you need to make your question clearer / more specifically about that. I would be really interested to know myself! $\endgroup$ – Resquiens Oct 27 '14 at 18:38
  • $\begingroup$ Yes, I understand that it gives the volume, but I ask for a formula that give me $z$, so that i can graph the solid in a grapher. Because as far I know (not much, to be honnest), The volume of the solid is independant from his position is the space (please explain me if I'm mistaking completely, I haven't done more calculus that derivative, even if I understand the integration and how it works and your demonstration, that is pretty clear btw) $\endgroup$ – Robin Carlier Oct 27 '14 at 18:48
  • $\begingroup$ I'll edit my question to be more clear, you are right $\endgroup$ – Robin Carlier Oct 27 '14 at 18:49

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