Clarifying the Volume of a Cone

[bodensee9]

I am wondering if someone can help clarify the following? Suppose that I’m asked to find the volume of a cone. So, the volume would be ∫∫∫dxdydz or using polar coordinates ∫∫∫rdzdrdθ. Therefore the volume would be if I have a cone with base radius R and height H, I can express the radius r at any height z using similar triangles. I would get that r = Rz/H, where z is the height at any point in the cone.

Hence the volume would be ∫∫∫rdzdrdθ where Rz/H ≤ r ≤ R, 0 ≤ z ≤ H, and 0 ≤ θ ≤ 2π.

But I’m wondering why I can’t express the volume as ∫∫∫rdzdrdθ where 0 ≤ r ≤ Rz/H, and 0 ≤ z ≤ H and 0 ≤ θ ≤ 2π? Thanks!

 

[quantumdude]

But I’m wondering why I can’t express the volume as ∫∫∫rdzdrdθ where 0 ≤ r ≤ Rz/H, and 0 ≤ z ≤ H and 0 ≤ θ ≤ 2π? Thanks!

Because you've got the bounds on r in terms of z. That means that the r integration has to come before the z integration.

 

[HallsofIvy]

I am wondering if someone can help clarify the following? Suppose that I’m asked to find the volume of a cone. So, the volume would be ∫∫∫dxdydz or using polar coordinates ∫∫∫rdzdrdθ. Therefore the volume would be if I have a cone with base radius R and height H, I can express the radius r at any height z using similar triangles. I would get that r = Rz/H, where z is the height at any point in the cone.

Hence the volume would be ∫∫∫rdzdrdθ where Rz/H ≤ r ≤ R, 0 ≤ z ≤ H, and 0 ≤ θ ≤ 2π.

No, it would not. The way you have set up this cone, it has vertex at the origin and base at z= H. The inside of the cone has 0≤ r≤ Rz/H. Your limits of integration gives the volume of the part of the cylinder of radius R and height H outside the cone.

But I’m wondering why I can’t express the volume as ∫∫∫rdzdrdθ where 0 ≤ r ≤ Rz/H, and 0 ≤ z ≤ H and 0 ≤ θ ≤ 2π? Thanks!

You can. This is the correct integral!
$$\int_{z=0}^H\int_{\theta= 0}^{2\pi}\int_{r= 0}^{Rz/H} rdrd\theta dz$$
$$= 2\pi \int_{z=0}^H (1/2)(R^2 z^2)/H^2 dz= \pi R^2/H^2 \int_0^H z^2$$
$$= \pi R^2/H^2 (1/3)H^3= (1/3)\pi R^2 H$$
which is the standard formula for area of a cone.

Your original formula,
$$\int_{z=0}^H\int_{\theta= 0}^{2\pi}\int_{r= Rz/H}^R rdrd\theta dz$$
$$= 2\pi \int_{z=0}^H (1/2)(R^2- R^2z^2/H^2)dz= \pi R^2\int_0^H(1- z^2/H^2)dz$$
$$= (2/3)\pi R^2 H$$
Notice that those two volumes add to [itex]\pi R^2 h[/tex], the volume of the cylinder.

(Tom, I think you misinterpreted his question. Well, one of us did!)

 

[bodensee9]

Thanks, that was helpful. But I'm still a little bit confused.
I am wondering how integrating from Rz/H ≤ r ≤ R gives you the volume outside of the cone? I understand that if you integrate from bottom up that would give you 0 ≤ r ≤ Rz/H. But, I thought that since you know that the maximum value for r is R, you can also write it as Rz/H ≤ r ≤ R?
Sorry.

 

[HallsofIvy]

Thanks, that was helpful. But I'm still a little bit confused.
I am wondering how integrating from Rz/H ≤ r ≤ R gives you the volume outside of the cone? I understand that if you integrate from bottom up that would give you 0 ≤ r ≤ Rz/H. But, I thought that since you know that the maximum value for r is R, you can also write it as Rz/H ≤ r ≤ R?
Sorry.

The axis of the cone is the positive z-axis, isn't it? So that r= 0 is always inside the cone, no matter what z is?

Draw a graph. Take the horizontal axis to be "r" and the vertical axis to be z. Draw the straight lines r= Rz/H or z= Hr/R and z= -Hr/R representing the sides of the cone and the vertical line r= R. Where is the inside of the cone and where is the outside? Where is Rz/H ≤ r ≤ R?

 

[bodensee9]

Okay, I see it now. Thanks so much!

 

[quantumdude]

(Tom, I think you misinterpreted his question. Well, one of us did!)

I confess that I didn't really read above the line that said, "why can't I express the volume as...?" in the OP. But my point is that he had the variables of integration out of order. dr has to be to the left of dz in the integrand to use the bounds that he stated.