Math Problem Statement
Question (4): Evaluate the following Integrals: 1) β y ππ π , where π is the βTetrahedronβ bounded by the planes: π₯ = 0, π¦ = 0, π§ = 0 and π₯ + π¦ + π§ = 1
Solution
To evaluate the integral , where is the tetrahedron bounded by the planes , , , and , we need to carefully set up the limits of integration and integrate the function over the volume of the tetrahedron.
Steps to Solve:
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Determine the Region :
- The tetrahedron is defined by the planes , , , and .
- The bounds for , , and can be found from these planes.
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Set Up the Integral:
- The plane implies that for fixed and , ranges from to .
- For fixed , ranges from to .
- ranges from to .
Thus, the integral can be written as:
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Evaluate the Innermost Integral (with respect to ):
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Evaluate the Second Integral (with respect to ): Evaluate each term separately: Combine these results:
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Evaluate the Outer Integral (with respect to ): Make a substitution , thus : Evaluate the integral: Therefore:
Final Answer:
Would you like further details or have any questions?
Here are some related questions for further practice:
- Evaluate for the same tetrahedron.
- Evaluate for the same tetrahedron.
- Find the volume of the tetrahedron using integration.
- Evaluate for the same tetrahedron.
- Evaluate for the same tetrahedron.
- Evaluate for the same tetrahedron.
- Find the centroid of the tetrahedron.
- Evaluate for the same tetrahedron.
Tip: When dealing with multiple integrals over a region bounded by planes, carefully determine the limits of integration by analyzing the bounding surfaces.
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Math Problem Analysis
Mathematical Concepts
Triple Integrals
Tetrahedron
Volume Calculation
Formulas
Triple integral setup
Volume of a tetrahedron
Theorems
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Suitable Grade Level
Advanced Mathematics
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