Finding the volume of the solid

In summary, to find the volume of the solid bounded by the three coordinate planes, bounded above by the plane x + y + z = 2, and bounded below by the z = x + y, you can use the equation ## \displaystyle\int_0^1\displaystyle\int_0^{1-x} (2-2x-2y) dy dx = \frac13 ##. However, be careful to correctly set the integration limits and consider the vertices of the upper and lower tetrahedrons in the first octant.
  • #1
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Homework Statement
Find the volume V of the solid S bounded by the three coordinate planes, bounded above by the plane x + y + z = 2, and bounded below by the z = x + y.
Relevant Equations
No equation
My attempt : ## \displaystyle\int_0^2 \displaystyle\int_0^{2-x} (2-2x -2y) dy dx = -\frac43 ## But it is wrong.
 
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  • #2
WMDhamnekar said:
Homework Statement:: Find the volume V of the solid S bounded by the three coordinate planes, bounded above by the plane x + y + z = 2, and bounded below by the z = x + y.
Relevant Equations:: No equation

My attempt : ## \displaystyle\int_0^2 \displaystyle\int_0^{2-x} (2-2x -2y) dy dx = -\frac43 ## But it is wrong.
Your first clue that you have done something wrong is that you got a negative number for the volume. Have you drawn a sketch of the solid? It looks like a pair of tetrahedrons, one on top of the other. At least one of your lower integration limits is wrong, as the base of the solid isn't in the x-y plane.
 
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  • #3
Mark44 said:
Your first clue that you have done something wrong is that you got a negative number for the volume. Have you drawn a sketch of the solid? It looks like a pair of tetrahedrons, one on top of the other. At least one of your lower integration limits is wrong, as the base of the solid isn't in the x-y plane.
My graphing calculator sketched this problem as follows:

1652714967358.png


1652714990008.png


1652715017648.png


Now, which volume, the author want to be computed by readers, viewers and/or students?
 
  • #4
The graphs from your calculator aren't very helpful. The sketch I drew on a piece of paper shows that the solid lies entirely in the first octant (the portion of space in which all three coordinates are nonnegative).

The vertices of the upper tetrahedron are at (0, 0, 2), (1, 0, 1), (0, 1, 1), and (0, 0, 1). Notice that the first three of these vertices lie on the plane z = 2 - x - y. The vertices of the lower tetrahedron are at (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1). The first three of these vertices lie on the plane z = x + y.
 
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  • #5
Mark44 said:
The graphs from your calculator aren't very helpful. The sketch I drew on a piece of paper shows that the solid lies entirely in the first octant (the portion of space in which all three coordinates are nonnegative).

The vertices of the upper tetrahedron are at (0, 0, 2), (1, 0, 1), (0, 1, 1), and (0, 0, 1). Notice that the first three of these vertices lie on the plane z = 2 - x - y. The vertices of the lower tetrahedron are at (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1). The first three of these vertices lie on the plane z = x + y.
Hi,
Is the following sketch as per your vertices correct?
solidgraph.png
 
  • #6
WMDhamnekar said:
Hi,
Is the following sketch as per your vertices correct?View attachment 301525
Yes, that looks like what I did. I haven't worked the problem, but I think you'll need the equation of the line that goes from (1, 0, 1) over to (0, 1, 1). This should be x + y = 1, in the plane z = 1.
 
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  • #7
Mark44 said:
Yes, that looks like what I did. I haven't worked the problem, but I think you'll need the equation of the line that goes from (1, 0, 1) over to (0, 1, 1). This should be x + y = 1, in the plane z = 1.
Sothe final answer to this question is ## \displaystyle\int_0^1\displaystyle\int_0^{1-x} (2-2x-2y) dy dx = \frac13 ##
 

1. How do you find the volume of a solid?

To find the volume of a solid, you must first measure the length, width, and height of the solid. Then, you can use the formula V = lwh (volume = length x width x height) to calculate the volume in cubic units.

2. Can the volume of a solid be negative?

No, the volume of a solid cannot be negative. Volume is a measure of the amount of space an object takes up, so it must always be a positive value.

3. What units are typically used to measure volume?

The most common units used to measure volume are cubic units, such as cubic centimeters, cubic feet, or cubic meters. However, other units such as liters or gallons can also be used depending on the context.

4. How is the volume of an irregularly shaped solid calculated?

If a solid has an irregular shape, you can still find its volume by using the displacement method. This involves submerging the solid in a container of water and measuring the amount of water displaced. The volume of the solid will be equal to the volume of water displaced.

5. Can the volume of a solid change?

Yes, the volume of a solid can change if its dimensions or shape change. For example, if you cut a solid in half, its volume will be reduced by half as well. However, the volume of a solid remains constant as long as its dimensions and shape remain the same.

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