What is the general formula for the volume of rotated polar curve?
Is it just the integral of pi*radius^2 ds,
where ds=sqrt(r^2+(dr/dtheta)^2)?
if r=theta is rotated about the x-axis,
why is ds=cos(theta)-r*sin(theta)?
I'm in a really hurry,
thank you,
Is it just the integral of pi*radius^2 ds,
where ds=sqrt(r^2+(dr/dtheta)^2)?
if r=theta is rotated about the x-axis,
why is ds=cos(theta)-r*sin(theta)?
I'm in a really hurry,
thank you,
-
In your (first) expression for ds, you neglected to multiply the RHS through by dθ. A differential can't equal an expression that is not a differential.
ds^2 = dr^2 + r^2 dθ^2
ds = √( r^2 + (dr/dθ)^2) dθ
But I don't see why you would need this to do a volume integral.
Finding volumes of polar curves that are rotated (about the x-axis, I presume?) can be a bit tricky, because if you try to convert everything to Cartesian (rectangular), you might wind up with a curve that turns back on itself in rectangular coordinates. The "safe" way to do this is by conical shells with their apices at the origin. The volume element is then:
dV = (2π/3) r^3 sinθ dθ
[If you want, I can try to set down here how I arrived at that. In any case, you can check it by letting
r = a,
and integrating from 0 to π. This is just a sphere of radius a. You can also check it for the line
x = a,
which in polar, is
r = a/cosθ,
integrating from 0 to θ1. This is a cone of height a, and radius a/cos(θ1).]
I am mystified by your last equation, ds = cosθ - r sinθ. Here again, there needs to be a factor of some kind of differential on the RHS. It also needs a factor having units of length in the first term on the right.
ds^2 = dr^2 + r^2 dθ^2
ds = √( r^2 + (dr/dθ)^2) dθ
But I don't see why you would need this to do a volume integral.
Finding volumes of polar curves that are rotated (about the x-axis, I presume?) can be a bit tricky, because if you try to convert everything to Cartesian (rectangular), you might wind up with a curve that turns back on itself in rectangular coordinates. The "safe" way to do this is by conical shells with their apices at the origin. The volume element is then:
dV = (2π/3) r^3 sinθ dθ
[If you want, I can try to set down here how I arrived at that. In any case, you can check it by letting
r = a,
and integrating from 0 to π. This is just a sphere of radius a. You can also check it for the line
x = a,
which in polar, is
r = a/cosθ,
integrating from 0 to θ1. This is a cone of height a, and radius a/cos(θ1).]
I am mystified by your last equation, ds = cosθ - r sinθ. Here again, there needs to be a factor of some kind of differential on the RHS. It also needs a factor having units of length in the first term on the right.