Finding Volume between Two planes Help

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Homework Help Overview

The discussion revolves around finding the volume of the region between the planes defined by the equations x+y+2z=2 and 2x+2y+z=4, specifically in the first quadrant using rectangular coordinates. Participants are exploring the setup for integrating to find this volume.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster graphed the planes and calculated intercepts, considering how to set up the integrals. They are uncertain about whether to subtract the functions representing the planes or to set limits for integration. Other participants suggest using double integrals and treating the problem as two separate mini-problems, while one participant questions the need for a triple integral.

Discussion Status

The discussion is ongoing, with participants providing different perspectives on how to approach the integration. Some guidance has been offered regarding the use of double integrals and the potential to subtract results, but there is no explicit consensus on the best method to proceed.

Contextual Notes

There is some confusion regarding the terminology used, with a participant correcting the original poster's reference to the "first quadrant" to "first octant," indicating a need for clarity in the spatial context of the problem. Additionally, there are differing opinions on whether to perform a triple integral or to handle the planes separately.

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Finding Volume between Two planes "Help"

Ok here's the question
Find the volume of the region between places x+y+2z=2 and 2x+2y+z=4 in THE FIRST QUADRANT, using rectangular coordinates.

What I have done:

Graphed the planes. Created x=o y=o and z=o planes to remain in first quadrant for my own visuals. I set both the planes equal to zero. Solved for my x y and z intercepts. x=2, y=2 for both and z=1 for one and z=4 for another.
Now the set up my integrals. I need to integrate the z to go between the two planes so I was thinking to subtract one from the other..? That or I make me lower limit one of the planes and the other my upper limit. This is what's hanging me up... whether to subtract or not. If it do then its not the same function , but if I don't then I think my limits will have no "limit" to go from z=0 to another limit.

Help
 
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Everyones lookin and nobodys saying anything. :/
 
One reason for that could be that this isn't the Homework Help section. Allow me to move your thread for you.


*kick*

Ah, there we are.

Ok here's the question
Find the volume of the region between places x+y+2z=2 and 2x+2y+z=4 in THE FIRST QUADRANT, using rectangular coordinates.

You mean first octant, don't you? This is 3-space, not 2-space.

What I have done:
.
.
.
so I was thinking to subtract one from the other..?

Go with that. Find the volume between each given plane and the xy-plane and subtract the results. You'll need to do a double integral in both cases, but you can treat them as 2 separate mini-problems.
 
Im supposed to do a triple integral. Are you saying I should do for dz a integral from O to lower plane - a integral from o to highest... Should I compute for y and x for each of those integrals separately and subtract two different answers?
 
well i get the same answer taking the integral of the lower plane completely and integral of upper plane completely, setting my dy=2-x both times, subtracting two answer= 2.

then i did a integral subtracting both the planes, y=2-x, x=0..2 and got 2 again.
 
Last edited:

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