Volume of a quarter cylinder between 2 planes

In summary, the conversation discussed finding the volume of a region within a quarter cylinder, bounded by certain conditions and given in cylindrical polar coordinates. The solution involved considering the volume of the cylinder up to a certain height, as well as the volume between two planes, and integrating with respect to the appropriate variable. The final result was π-2/3.
  • #1
Matternot
Gold Member
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Homework Statement


Find the volume V of the region that lies inside the quarter cylinder
0 ≤ r ≤ 1, 0 ≤ θ ⇐ 1/2 π and between the planes x+y+z=4 and z=0, where (r, θ, z) are cylindrical polar coordinates.

Homework Equations


integral dV = integral r drdθdz

The Attempt at a Solution


I considered 4pi (volume of cylinder up to z=4) and the volume between the planes z=0 and x+y+z = 0

My biggest problem is working out which I should take as my final variable to integrate. I assume it'll be r
 
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  • #2
Oops... Came up with a solution leading to the result π-2/3 immediately after posting. If anyone wants to check my answer, be my guest.
 
Last edited:
  • #3
Stephen Hodgson said:
Oops... Came up with a solution leading to the result π-2/3 immediately after posting. If anyone wants to check my answer, be my guest.
Looks right.
 
  • #4
haruspex said:
Looks right.

Thanks
 

1. How do you calculate the volume of a quarter cylinder between 2 planes?

The volume of a quarter cylinder between 2 planes can be calculated by first finding the volume of the entire cylinder and then subtracting the volume of the missing portion. The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. To find the missing volume, you can use the formula for the volume of a cone, V = (1/3)πr^2h, where r is the radius and h is the height of the missing portion. Once you have both volumes, you can subtract the volume of the cone from the volume of the cylinder to find the volume of the quarter cylinder between 2 planes.

2. Can the quarter cylinder between 2 planes have a negative volume?

No, the volume of the quarter cylinder between 2 planes cannot be negative. Volume is a measure of the amount of space occupied by an object and cannot be negative. If the calculated volume is negative, it may indicate an error in the calculation or that the planes are not placed correctly.

3. How do you determine the height of the missing portion in a quarter cylinder between 2 planes?

The height of the missing portion in a quarter cylinder between 2 planes can be determined by subtracting the height of the cylinder from the height of the entire object. This can be done by measuring the heights of both planes and subtracting them from each other. Alternatively, if the height of the cylinder is known, you can simply subtract it from the total height of the object.

4. Can the quarter cylinder between 2 planes have a volume of zero?

Yes, the quarter cylinder between 2 planes can have a volume of zero if the planes are parallel and there is no missing portion. In this case, the object would be a full cylinder and the volume would be calculated using the formula V = πr^2h, where r is the radius and h is the height of the cylinder.

5. How do you convert the volume of a quarter cylinder between 2 planes to different units?

To convert the volume of a quarter cylinder between 2 planes to different units, you can use a conversion factor. For example, if the calculated volume is in cubic meters and you want to convert it to cubic feet, you can use the conversion factor 1 cubic meter = 35.3147 cubic feet. Multiply the volume in cubic meters by this conversion factor to get the volume in cubic feet. Alternatively, you can use an online unit converter to easily convert the volume to different units.

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