Find Volume of Region: Tetrahedron Bounded by Coordinate Planes & Plane

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In summary, The problem involves finding the volume of a region that is a tetrahedron in the first octant, bounded by the coordinate planes and a plane passing through (1, 0, 0), (0, 2, 0), and (0, 0, 3). The correct solution requires finding the equations of the planes and solving a triple integral with the appropriate limits of integration. The answer is 1, not 6 as initially thought.
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DWill
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Find the volume of this region: The tetrahedron in the first octant bounded by the coordinate planes and the plane passing through (1, 0, 0), (0, 2, 0), and (0, 0, 3).


Looking at this problem I thought it just involved solving a fairly simple triple integral:

||| dz dy dx

With these limits of integration:
0 <= x <= 1
0 <= y <= 2
0 <= z <= 3

I get the answer 6, but my textbook says the answer is 1. Is this a typo in the textbook or did I do something stupid?
 
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  • #2
What are the equations of the planes which make up the tetrahedron? What you have done appears to be calculating the volume of a cuboid in the first octant of dimensions 1x2x3. That's not the shape of a tetrahedron.
 
  • #3
Oh I see, I have to come up with the equations of the planes myself. that was stupid of me..thanks!
 
  • #4
Using constants for limits of integration gives the volume of the rectangular solid 0 <= x <= 1, 0 <= y <= 2, 0 <= z <= 3.
 

FAQ: Find Volume of Region: Tetrahedron Bounded by Coordinate Planes & Plane

What is a tetrahedron?

A tetrahedron is a three-dimensional shape with four triangular faces, six edges and four vertices. It is a type of pyramid and is one of the five platonic solids.

How do you find the volume of a tetrahedron?

The volume of a tetrahedron can be calculated using the formula V = (1/3) * b * h, where b is the area of the base and h is the height of the tetrahedron. Alternatively, it can also be calculated using the formula V = (1/3) * a^3, where a is the length of one of the edges.

What are coordinate planes?

Coordinate planes are two-dimensional planes used to locate points in a three-dimensional space. They are typically labeled as x, y, and z and are used in mathematical equations and graphs.

How do the coordinate planes and plane intersect to form the region of the tetrahedron?

The coordinate planes and the plane intersect at their respective axes (x, y, z) to form the boundaries of the tetrahedron. The coordinate planes form the base of the tetrahedron while the plane forms one of its faces.

Can you find the volume of a tetrahedron if it is not bounded by the coordinate planes and plane?

Yes, the volume can still be calculated using the same formula mentioned in question 2. The only difference would be that the base and height of the tetrahedron would have to be measured or calculated differently, depending on the given information.

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