Volume of a quarter cylinder between 2 planes

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Homework Help Overview

The problem involves finding the volume of a region within a quarter cylinder defined by specific cylindrical polar coordinates and bounded by two planes. The subject area relates to calculus and volume integration in three-dimensional space.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to determine the appropriate variable for integration and considers the volume of the cylinder up to a certain height. Some participants question the validity of the approach and the final variable choice.

Discussion Status

There has been some progress, with the original poster arriving at a potential solution. However, the discussion includes requests for verification of the result, indicating that multiple interpretations or confirmations are being explored.

Contextual Notes

Participants are working under the constraints of the problem statement and the definitions of the cylindrical coordinates. There is an indication of uncertainty regarding the integration process and the setup of the problem.

Matternot
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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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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:
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.
 
haruspex said:
Looks right.

Thanks
 

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