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.

The field is F⇀=zk⇀ and the portion of the paraboloid z=x2+y2 bounded above z=4. The parametric…

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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Question

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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