By using either a double or triple integral, find the volume of the tetrahedron with vertices (0,0,0), (1,0,0), (0, 1, 0), and (0, 0, 1).
By using either a double or triple integral, find the volume of the tetrahedron with vertices (0,0,0), (1,0,0), (0, 1, 0), and (0, 0, 1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
By using either a double or triple integral, find the volume of the tetrahedron with vertices \((0, 0, 0)\), \((1, 0, 0)\), \((0, 1, 0)\), and \((0, 0, 1)\).
**Explanation:**
To solve this problem, consider using a triple integral. The tetrahedron is bounded by the planes:
1. \(x = 0\)
2. \(y = 0\)
3. \(z = 0\)
4. \(x + y + z = 1\)
The integration limits for \(x\), \(y\), and \(z\) are thus determined by these boundaries. The volume \(V\) can be found using the integral:
\[ V = \int_{0}^{1} \int_{0}^{1-x} \int_{0}^{1-x-y} \, dz \, dy \, dx \]
Solving this integral will yield the volume of the tetrahedron.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fd531f0-250f-490d-993a-3449a32c15bc%2F68e7b100-72a5-4ff9-9963-e06bb0750c2c%2Fnrqaw4p_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
By using either a double or triple integral, find the volume of the tetrahedron with vertices \((0, 0, 0)\), \((1, 0, 0)\), \((0, 1, 0)\), and \((0, 0, 1)\).
**Explanation:**
To solve this problem, consider using a triple integral. The tetrahedron is bounded by the planes:
1. \(x = 0\)
2. \(y = 0\)
3. \(z = 0\)
4. \(x + y + z = 1\)
The integration limits for \(x\), \(y\), and \(z\) are thus determined by these boundaries. The volume \(V\) can be found using the integral:
\[ V = \int_{0}^{1} \int_{0}^{1-x} \int_{0}^{1-x-y} \, dz \, dy \, dx \]
Solving this integral will yield the volume of the tetrahedron.
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