Volume for a cone in cylindrical coordinates.

In summary, the conversation discusses using iterated integrals to find the volume of a cone using cylindrical coordinates. The integral is set up correctly, but the limits may be incorrect. The correct formula for the volume of a cone is given, and the conversation concludes with a humorous remark about the variable of integration.
  • #1
Telemachus
835
30

Homework Statement


Hi there. I haven't used iterated integrals for a while, and I'm studying some mechanics, the inertia tensor, etc. so I need to use some calculus. And I'm having some trouble with it.

I was trying to find the volume of a cone, and then I've found lots of trouble with such a simple problem.

So I thought of using cylindrical coordinates this way:
[tex]\begin{Bmatrix}{ x=r\cos\theta} \\y=r\sin\theta \\z=r\end{matrix}[/tex]

And then I've stated the integral this way:

[tex]\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{r}\displaystyle\int_{r}^{h}rdzdrd\theta=\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{r}r(h-r)drd\theta=\displaystyle\int_{0}^{2\pi}\displaystyle\frac{r^2h}{2}-\displaystyle\frac{r^3}{3}=\pi r^2h-\displaystyle\frac{2\pi\r^3}{3}=\pi r^2(h-\displaystyle\frac{2}{3}r)[/tex]

But I should get: [tex]V_{cone}=\displaystyle\frac{\pi r^2 h}{3}[/tex]

I think I'm giving wrong limits for the integration.

Help pls :)
 
Last edited:
Physics news on Phys.org
  • #2
Hi Telemachus! :smile:
Telemachus said:
I think I'm giving wrong limits for the integration.

No, your integration is fine. :smile:

(I'd have used spherical coordinates, but your way does work)

But you're doing it for the 45° cone instead of a general cone (so r = h, which makes your formula the same as the given answer).

Somehow your variable of integration r has managed to survive into the afterlife under a new persona. :wink:
 
  • #3
Thanks Tim :)

Haha sorry for the notation, I should used another name for the variable :P
 

FAQ: Volume for a cone in cylindrical coordinates.

1. What is the formula for calculating the volume of a cone in cylindrical coordinates?

The formula for calculating the volume of a cone in cylindrical coordinates is V = (1/3)πr2h, where r is the radius of the base and h is the height of the cone.

2. How do you convert a cone from Cartesian coordinates to cylindrical coordinates?

To convert a cone from Cartesian coordinates (x, y, z) to cylindrical coordinates (r, θ, z), use the equations r = √(x2 + y2) and θ = tan-1(y/x) to find the polar coordinates of the cone's base. The z-coordinate remains the same.

3. Can you find the volume of a cone in cylindrical coordinates if the height is not given?

No, the height is a necessary component in calculating the volume of a cone in cylindrical coordinates. Without the height, the formula V = (1/3)πr2h cannot be used to find the volume.

4. Is it possible to use the same formula for finding the volume of a cone in both Cartesian and cylindrical coordinates?

Yes, the formula for calculating the volume of a cone (V = (1/3)πr2h) can be used in both Cartesian and cylindrical coordinates. However, the values for r and h will be different depending on the coordinate system being used.

5. How does the volume of a cone in cylindrical coordinates compare to the volume of a cone in Cartesian coordinates?

The volume of a cone in cylindrical coordinates will be the same as the volume of a cone in Cartesian coordinates if the cone has a circular base. However, if the base is not circular, then the volumes will be different due to the different formulas used to calculate them.

Back
Top