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5. Find the volume of the region bounded by the paraboloids z = 12 – x² - y² and z = 2x² + 2y². 2

5. Find the volume of the region bounded by the paraboloids z = 12 – x² - y² and z = 2x² + 2y². 2

Advanced Engineering Mathematics
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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Use Cylindrical Coordinates 

**Problem 5: Volume Bounded by Paraboloids** Find the volume of the region bounded by the paraboloids \( z = 12 - x^2 - y^2 \) and \( z = 2x^2 + 2y^2 \). **Explanation:** This mathematical problem asks for the calculation of the volume enclosed between two three-dimensional surfaces represented by paraboloid equations. 1. **Equation Descriptions:** - The first equation \( z = 12 - x^2 - y^2 \) represents an inverted paraboloid, which opens downwards. The vertex of this paraboloid is located at the point \( (0, 0, 12) \). - The second equation \( z = 2x^2 + 2y^2 \) represents a standard upward-opening paraboloid with its vertex at the origin \( (0, 0, 0) \). 2. **Graphical Interpretation:** - If this were plotted graphically, the intersection of these two parabolic surfaces would form a closed, bounded region in the space. The task is to determine the volume contained within this intersection. **Approach:** To find the volume of the region, one could set up a triple integral in cylindrical coordinates and integrate over the appropriate region of intersection. Alternatively, using symmetry or computational tools would also yield the solution to this problem efficiently.
Transcribed Image Text:**Problem 5: Volume Bounded by Paraboloids** Find the volume of the region bounded by the paraboloids \( z = 12 - x^2 - y^2 \) and \( z = 2x^2 + 2y^2 \). **Explanation:** This mathematical problem asks for the calculation of the volume enclosed between two three-dimensional surfaces represented by paraboloid equations. 1. **Equation Descriptions:** - The first equation \( z = 12 - x^2 - y^2 \) represents an inverted paraboloid, which opens downwards. The vertex of this paraboloid is located at the point \( (0, 0, 12) \). - The second equation \( z = 2x^2 + 2y^2 \) represents a standard upward-opening paraboloid with its vertex at the origin \( (0, 0, 0) \). 2. **Graphical Interpretation:** - If this were plotted graphically, the intersection of these two parabolic surfaces would form a closed, bounded region in the space. The task is to determine the volume contained within this intersection. **Approach:** To find the volume of the region, one could set up a triple integral in cylindrical coordinates and integrate over the appropriate region of intersection. Alternatively, using symmetry or computational tools would also yield the solution to this problem efficiently.
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