Find the volume of the solid S that is bounded by the elliptic paraboloid x² + 2y2 + z = 32, the planes x = 2 and y = 2, and the three coordinate planes.
Find the volume of the solid S that is bounded by the elliptic paraboloid x² + 2y2 + z = 32, the planes x = 2 and y = 2, and the three coordinate planes.
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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![Find the volume of the solid S that is bounded by the elliptic paraboloid x² + 2y²
Solution
V =
=
= [² / ² (3²-
We first observe that S is the solid that lies under the surface z = 32 - x² - 2y² and above the square R = [0, 2] × [0, 2]. (See the figure above.) We are now in a position to evaluate the double integral using Fubini's Theorem. Therefore we have the following.
(32 - x² – 2y²) dA
= 1/₁²
=
x
32x-
Interactive 3D Graph
Graph
184
=
· 6² ( ¹34 - [
1²
107
30
-207
30
20
FI
(32 - x² – 2y²) dx dy
10
Graph Description
dy
1x = 2
- 2y²xx
¹x = 0
Help
+ z = 32, the planes x = 2 and y = 2, and the three coordinate planes.
dy](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2acb91cb-06a2-4f4a-b63b-95f3aaaf3e2a%2F18d0737d-b1d1-49e9-9595-6d327342ab99%2Fme4gj65_processed.png&w=3840&q=75)
Transcribed Image Text:Find the volume of the solid S that is bounded by the elliptic paraboloid x² + 2y²
Solution
V =
=
= [² / ² (3²-
We first observe that S is the solid that lies under the surface z = 32 - x² - 2y² and above the square R = [0, 2] × [0, 2]. (See the figure above.) We are now in a position to evaluate the double integral using Fubini's Theorem. Therefore we have the following.
(32 - x² – 2y²) dA
= 1/₁²
=
x
32x-
Interactive 3D Graph
Graph
184
=
· 6² ( ¹34 - [
1²
107
30
-207
30
20
FI
(32 - x² – 2y²) dx dy
10
Graph Description
dy
1x = 2
- 2y²xx
¹x = 0
Help
+ z = 32, the planes x = 2 and y = 2, and the three coordinate planes.
dy
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