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Using cylindrical coordinates, find the volume in space that is enclosed by the parabolid z = 2 + x² + y² and the plane z = 6 in the first octant.

Using cylindrical coordinates, find the volume in space that is enclosed by the parabolid z = 2 + x² + y² and the plane z = 6 in the first octant.

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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Using cylindrical coordinates, find the volume in space that is enclosed by the parabolid z = 2+ x² + y² and the plane z = 6 in the first octant.
Transcribed Image Text:Using cylindrical coordinates, find the volume in space that is enclosed by the parabolid z = 2+ x² + y² and the plane z = 6 in the first octant.
Expert Solution
Solution:

The paraboloid is z=2+x2+y2 and the plane is z=6.

2+x2+y2=6x2+y2=4r2=4r=±2

Thus, the limits of integration in the first octant is  0≤r≤2, 2+x2+y2≤z≤6, 0≤θ≤π2.

Using cylindrical coordinates: r2=x2+y2, dxdy=rdr dθ

Thus, the triple integral becomes

∭dV=∫0π2∫02∫2+r26rdzdrdθ

 

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