   Chapter 16.9, Problem 12E

Chapter
Section
Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∫∫s F · dS; that is, calculate the flux of F across S.12. F(x, y, z) = (xy + 2xz) i + (x2 + y2) j + (xy - z2) k, S is the surface of the solid bounded by the cylinder x2 + y2 = 4 and the planes z = y - 2 and z = 0

To determine

To calculate: The flux of vector field F(x,y,z)=(xy+2xz)i+(x2+y2)j+(xyz2)k across the surface of the solid S which is bounded by the cylinder x2+y2=4 , and the planes z=y2 and z=0 .

Explanation

Given data:

The vector field is F(x,y,z)=(xy+2xz)i+(x2+y2)j+(xyz2)k .

The surface of solid S is bounded by the cylinder x2+y2=4 , and the planes z=y2 and z=0 .

Formula used:

Write the expression to find flux of the vector field F(x,y,z) across the surface S .

SFdS=EdivFdV (1)

Here,

E is the solid region.

Write the expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk .

divF=xP+yQ+zR (2)

Write the equation of cylinder as follows.

x2+y2=4

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

x=rcosθ,y=rsinθ,0r2,0θ2π

As the surface S is bounded by the cylinder x2+y2=4 , and the planes z=y2 and z=0 , the limits of z, r, and θ are written as follows.

y2z00r20θ2π

Calculation of divF :

Substitute (xy+2xz) for P , (x2+y2) for Q , and (xyz2) for R in equation (2),

divF=x(xy+2xz)+y(x2+y2)+z(xyz2)=(y+2z)+(0+2y)+(02z)=y+2z+2y2z=3y

Calculation of flux of vector field:

Substitute 3y for divF in equation (1),

SFdS=E(3y)dV

Apply the limits of bounded region of surface, substitute rsinθ for y , and rewrite the expression as follows.

SFdS=02π02rsinθ20(3rsinθ)dz(rdr)dθ=02π023r2sinθdrdθrsinθ20(1)dz=02π023r2sinθdrdθ[z]rsinθ20=02π023r2sinθdrdθ[0(rsinθ2)]

Simplify the expression as follows

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 2336, find the domain of the function. 28. f(x)=x5

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Problems 1-4, graph the functions. 4.

Mathematical Applications for the Management, Life, and Social Sciences

Express the number as a ratio of integers. 55. 1.234567

Single Variable Calculus: Early Transcendentals

True or False: If f′(x) = g′(x), then f(x) = g(x).

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: converges if and only if .

Study Guide for Stewart's Multivariable Calculus, 8th 