Let S Be the Surface of the Upper Part of the Cylinder

question 8

Essay

Let S be the surface of the upper part of the cylinder 4x² + z² = 1, z $\ge$ 0, between the planes y = -1, y = 1, with an upward-pointing normal.
(a)Evaluate the flux integral $\int _ { S } \left( - 3 x y ^ { 2 } z \vec { i } + 7 \cos z \vec { j } + 3 \vec { k } \right) \cdot \vec { d A }$ (b)Consider W, the solid region described by -1 $\le$ y $\le$ 1, 4x² + z² $\le$ 1, z $\ge$ 0.Evaluate $\int _ { W } \operatorname { div } \left( - 3 x y ^ { 2 } z \vec { i } + 7 \cos z \vec { j } + 3 \vec { k } \right) d V$ Does this contradict the Divergence Theorem? Explain.

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Definitions:

Ordered Pairs

A pair of elements (x, y), representing coordinates on a Cartesian plane.

Equation

An assertion in mathematics indicating that two expressions, often incorporating variables, are equivalent.

Line Segments

A line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

Points

Locations in space defined by their coordinates on a graph or geometric context.