   Chapter 16, Problem 34RE

Chapter
Section
Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∬S F · dS, where F(x, y, z) = x3 i + y3 j + z3 k and S is the surface of the solid bounded by the cylinder x2 + y2 = 1 and the planes z = 0 and z = 2.

To determine

To calculate: The surface integral SFdS for F(x,y,z)=x3i+y3j+z3k across the surface of the solid S, which is bounded by the cylinder x2+y2=1 , and the planes z=0 and z=2 .

Explanation

Given data:

The vector field is F(x,y,z)=x3i+y3j+z3k .

The surface of solid S is bounded by the cylinder x2+y2=1 , and the planes z=0 and z=2 .

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=1

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

x=rcosθ,y=rsinθ,0r1,0θ2π

As the surface S is bounded by the cylinder x2+y2=1 , and the planes z=0 and z=2 , the limits of x, r, and θ are written as follows.

0z20r10θ2π

Find the value of divF .

Substitute x3 for P , y3 for Q , and z3 for R in equation (2),

divF=x(x3)+y(y3)+z(z3)=3x2+3y2+3z2=3(x2+y2+z2)

Substitute rcosθ for x and rsinθ for y ,

divF=3((rcosθ)2+(rsinθ)2+z2)=3(r2cos2θ+r2sin2θ+z2)=3(r2+z2)

Substitute 3(r2+z2) for divF in equation (1),

SFdS=E3(r2+z2)dV

Apply the limits of bounded region of surface and rewrite the expression as follows

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