Chapter 16.9, Problem 7E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∫∫_{s}**F** · *d*S; that is, calculate the flux of **F** across *S.*

**7. F**(*x*, *y*, *z*) = 3*xy*^{2} **i** + *xe ^{z}*

To determine

**To calculate:** The flux of vector field
*S*, which is bounded by the cylinder

Explanation

**Given data:**

The vector field is

The surface of solid

**Formula used:**

Write the expression to find flux of the vector field

Here,

Write the expression to find divergence of vector field

Write the equation of cylinder as follows.

From the equation of cylinder, the parameters are considered as follows.

As the surface
*x*, *r*, and

**Calculation of**
**:**

Substitute

**Calculation of flux of vector field:**

Substitute

Apply the limits of bounded region of surface and substitute

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