x2 + y? that lies below the plane z = 4 (see diagram above), Let F = zkand let S be the portion of the paraboloid z = oriented upward. Find C. F. dÃ. Please give your answer as a decimal rounded to the nearest 0.001.
x2 + y? that lies below the plane z = 4 (see diagram above), Let F = zkand let S be the portion of the paraboloid z = oriented upward. Find C. F. dÃ. Please give your answer as a decimal rounded to the nearest 0.001.
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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Transcribed Image Text:Let \(\vec{F} = z \hat{k}\), and let \(S\) be the portion of the paraboloid \(z = x^2 + y^2\) that lies below the plane \(z = 4\) (see diagram above), oriented upward. Find \(\iint_S \vec{F} \cdot d\vec{A}\). Please give your answer as a decimal rounded to the nearest 0.001.
**Diagram Explanation:**
The diagram shows a 3D graph of a paraboloid, which is shaped like an upward-opening bowl. The axes are labeled \(x\), \(y\), and \(z\). The surface in question is part of this paraboloid, specifically where \(z\) ranges from \(0\) up to \(4\), forming a cap on top. The field vector \(\vec{F}\) is oriented in the \(z\)-direction.
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