Finding Volume between Two planes Help

In summary, the task is to find the volume between two planes (x+y+2z=2 and 2x+2y+z=4) in the first quadrant using rectangular coordinates. The solution involves graphing the planes, finding the x, y, and z intercepts, and setting up integrals. The question of whether to subtract or not when integrating between the two planes is addressed and it is recommended to subtract the volume between each plane and the xy-plane. The solution involves a triple integral and finding the volume between each plane and the xy-plane separately, then subtracting the results. Ultimately, the final answer is 2.
  • #1
CaityAnn
38
0
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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  • #2
Everyones lookin and nobodys saying anything. :/
 
  • #3
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.
 
  • #4
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 seperately and subtract two different answers?
 
  • #5
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:

What is the definition of "volume between two planes"?

The volume between two planes refers to the amount of space that is enclosed or bounded by the two planes. It is a three-dimensional measurement that can be calculated by finding the area of the cross-section formed by the two planes and then multiplying it by the distance between the planes.

What is the formula for finding the volume between two planes?

The formula for finding the volume between two planes is V = A * h, where V is the volume, A is the area of the cross-section, and h is the distance between the planes. This formula assumes that the two planes are parallel to each other.

How do you find the area of the cross-section between two planes?

The area of the cross-section between two planes can be found by finding the intersection line between the two planes and then using the appropriate formula for the shape of the cross-section. For example, if the cross-section is a rectangle, the area can be found by multiplying the length and width of the rectangle.

What are some real-world applications of finding volume between two planes?

Finding the volume between two planes is a useful concept in fields such as architecture, engineering, and construction. It can be used to determine the amount of material needed to fill a space, such as concrete for a foundation or water for a swimming pool. It is also used in calculating the volume of liquids in containers with non-parallel sides.

What are some common mistakes when finding volume between two planes?

Some common mistakes when finding volume between two planes include using the wrong formula, not considering the units of measurement, and not accounting for irregular cross-sections. It is important to carefully review the problem and choose the appropriate formula and units before calculating the volume. Additionally, accurately measuring and accounting for any irregularities in the cross-section is crucial for an accurate result.

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