Triple Integral Volume: Octant x,y,z>=0 Bounded by x+y+1=1 and x+y+2z=1

In summary, the conversation discusses finding the volume of a solid in the first octant bounded by two planes, x+y+1=1 and x+y+2z=1. The attempt at a solution involves finding the projection of the solid onto the xy plane and determining the bounds for x and z. The conversation also mentions confusion about the lower bound for x and the bounds for y, and a suggested method for determining the volume using integration.
  • #1
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Homework Statement


Find the volume of the solid in the octant x,y,z>=0, bounded by x+y+1=1 and x+y+2z=1


The Attempt at a Solution


I've been looking at an example in the textbook that is similar to this problem. First, I found the projection of W onto the xy plane:

x+y-1=(x/2)+(y/2)-(1/2)

(x/2)+(y/2)=(1/2)

This let's you find the bounds of x in terms of y (The bounds of z are just the two functions above).

x=1-y

Here is where I got confused. In the book, with different functions, they used the same process. But the book had 0<=x<=(their function) and 0<=y<=1

and I'm not sure where the lower bound of x or either of the bounds of y came from.

If someone could either clarify where these came from and/or help me continue with my own problem, that would be great.
 
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  • #2
The plane x+y+1=1 does not go through the first octant. How about drawing them? I mean, I did a quick sketch then just confirmed it in Mathematica:

Code:
ContourPlot3D[{x + y + 1 == 1, 2 z == 1 - x - y},
 {x, -5, 5}, {y, -5,  5}, {z, -5, 5},
 PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}]

If you wanted just underneath the plane 2z=1-x-y in the first octant, then would it not be:

[tex]\int_0^1\int_0^{1-x} 1/2(x-y)dzdydz[/tex]
 

What is a triple integral volume?

A triple integral volume is a mathematical concept used in multivariable calculus to find the volume of a three-dimensional object or region. It involves integrating a function over three variables, typically representing the x, y, and z coordinates of a point in space.

How is a triple integral volume calculated?

To calculate a triple integral volume, you first need to define the boundaries of the region you want to find the volume of. Then, you set up the integral using the function that represents the object or region, and the appropriate limits of integration for each variable. Finally, you solve the integral using techniques such as substitution, integration by parts, or partial fractions.

What are some real-world applications of triple integral volume?

Triple integral volume has many real-world applications in fields such as physics, engineering, and economics. It can be used to calculate the volume of irregularly shaped objects, the mass of a three-dimensional object with varying density, the center of mass of a solid object, and the volume of a fluid in motion.

What are the limitations of triple integral volume?

One limitation of triple integral volume is that it can only be used for objects or regions with continuous functions. Additionally, it can be difficult to set up and solve the integral for complex objects or regions, and it may require advanced mathematical techniques.

How does triple integral volume relate to other types of integrals?

Triple integral volume is a specific type of multiple integral, which is a generalization of single integrals used to find the area under a curve. It is also related to line integrals, which are used to calculate work or circulation along a path. Triple integral volume is a higher-dimensional version of these integrals, used to find the volume of a three-dimensional object or region.

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