Volume of a cone using cylindrical coordinates and integration

In summary, the conversation discusses finding the volume of a cone using integration with cylindrical coordinates. The speaker initially sets up the integral as \int_{0}^{2\pi}\int_{0}^{h}\int_{0}^{R}\rho d\rho dz d\phi, but realizes that this is for a cylinder and not a cone. They then draw a picture and explain that the integral for the cone should go from 0 to h(1- r/R) in the z-direction. The final volume formula is V_{cone} = \frac{1}{3}\pi r^2 h.
  • #1
jolt527
8
0
Hi all! I was trying to figure out how to find the volume of a cone with radius R and height h using integration with cylindrical coordinates. I first tried to set the the integral as:

[tex]\int_{0}^{2\pi}\int_{0}^{h}\int_{0}^{R}\rho d\rho dz d\phi[/tex]

...but I think that this is setting up the integral for a cylinder and not a cone. Any suggestions so I end up with the correct volume?

[tex]V_{cone} = \frac{1}{3}\pi r^2 h[/tex]

Thank you!
 
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  • #2
Draw a picture. Assuming the cone has base of radius R and height h, in the xz-plane, it will be a triangle with vertices at (-R, 0), (R, 0), and (0, h). The slant side will be a line from (R,0) to (0,h) and that has equation z= h- (h/R)x= h(1- x/R). Rotating that around the z-axis to get a cone, "x" becomes "r": z= h(1- r/R). Your z integral should go from 0 to that.
 

Related to Volume of a cone using cylindrical coordinates and integration

1. What are cylindrical coordinates and how are they used in finding the volume of a cone?

Cylindrical coordinates are a type of coordinate system that is commonly used in three-dimensional space. They consist of a radial distance from a central point, an angle from a reference plane, and a height along a vertical axis. In finding the volume of a cone, cylindrical coordinates are used to express the cone's shape in terms of these coordinates, making it easier to integrate and calculate the volume.

2. What is the formula for finding the volume of a cone using cylindrical coordinates?

The formula for finding the volume of a cone using cylindrical coordinates is: V = ⅓r2hθ, where r is the radius of the base of the cone, h is the height of the cone, and θ is the angle of the cone's slope. This formula is derived from the general formula for finding the volume of a solid using integration.

3. How is integration used to find the volume of a cone using cylindrical coordinates?

Integration is used to find the volume of a cone using cylindrical coordinates by taking the limit of smaller and smaller slices of the cone, calculating the volume of each slice using the formula mentioned above, and then adding all of the volumes together. This process is known as a definite integral and is represented by the symbol ∫.

4. Can the volume of a cone be found using other coordinate systems?

Yes, the volume of a cone can also be found using other coordinate systems, such as Cartesian coordinates or spherical coordinates. However, using cylindrical coordinates is often the most convenient and efficient method for calculating the volume of a cone.

5. How is the volume of a cone using cylindrical coordinates related to the volume of a cylinder?

The volume of a cone using cylindrical coordinates is related to the volume of a cylinder in that both can be expressed using the same formula: V = πr2h. However, the cone's volume is only one-third of the cylinder's volume, as the cone is a third of the height of the cylinder. This relationship can also be seen in the formula for finding the volume of a cone using cylindrical coordinates, where the ⅓ represents the cone's proportion to the cylinder.

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