Find the volume of the indicated región. The region bounded by the Paraboloid z=1- y? 9 and the 16 xy-Plane
Find the volume of the indicated región. The region bounded by the Paraboloid z=1- y? 9 and the 16 xy-Plane
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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Question
![### Problem Statement
**Task:** Find the volume of the indicated region.
**Description:** The region is bounded by the paraboloid and the \(xy\)-plane.
**Equation of the Paraboloid:**
\[
z = 1 - \frac{x^2}{9} - \frac{y^2}{16}
\]
### Explanation
The equation describes a paraboloid that opens downward with its vertex at \((0, 0, 1)\). The goal is to find the volume of the region enclosed by this surface and the \(xy\)-plane, which is defined as the plane where \(z = 0\).
### Approach
1. **Understand the Region:** The surface is cut by the \(xy\)-plane. The region's boundary is determined by setting \(z = 0\).
2. **Volume Calculation:** The volume can be found by evaluating a double integral of the function over the region in the \(xy\)-plane.
3. **Double Integral Setup:** This involves integrating over the range of \(x\) and \(y\) values where the paraboloid is above the \(xy\)-plane.
Further calculations involve using polar coordinates or directly calculating the double integral in Cartesian coordinates, depending on the ease and symmetry observed in the function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12c62ea9-2423-4a35-a6cd-74646c6bbd41%2F1bfb55f7-9d42-438b-ac3a-c128f934646b%2Fr7xctc7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Task:** Find the volume of the indicated region.
**Description:** The region is bounded by the paraboloid and the \(xy\)-plane.
**Equation of the Paraboloid:**
\[
z = 1 - \frac{x^2}{9} - \frac{y^2}{16}
\]
### Explanation
The equation describes a paraboloid that opens downward with its vertex at \((0, 0, 1)\). The goal is to find the volume of the region enclosed by this surface and the \(xy\)-plane, which is defined as the plane where \(z = 0\).
### Approach
1. **Understand the Region:** The surface is cut by the \(xy\)-plane. The region's boundary is determined by setting \(z = 0\).
2. **Volume Calculation:** The volume can be found by evaluating a double integral of the function over the region in the \(xy\)-plane.
3. **Double Integral Setup:** This involves integrating over the range of \(x\) and \(y\) values where the paraboloid is above the \(xy\)-plane.
Further calculations involve using polar coordinates or directly calculating the double integral in Cartesian coordinates, depending on the ease and symmetry observed in the function.
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