   Chapter 16, Problem 30RE

Chapter
Section
Textbook Problem

Evaluate the surface integral.30. ∬S F · dS, where F(x, y, z) = x2 i − xy j + z k and S is the part of the paraboloid z = x2 + y2 below the plane z = 1 with upward orientation

To determine

To Evaluate: The surface integral of SFdS , where F(x,y,z)=x2i+xyj+zk

Explanation

Given data:

F(x,y,z)=x2i+xyj+zk , z=x2+y2 , z=1 .

Formula used:

Write the expression for surface integral.

SFdS=DF(ru×rv)dA (1)

Write the expression for rx×ry .

rx×ry=zxizyj+k (2)

Write the required differentiation formula.

ddx(x2)=2x

Consider the expression as follows.

z=x2+y2 (3)

Differentiate equation (3) with respect to x .

zx=2x

Differentiate equation (3) with respect to y .

zy=2y

Substitute 2x for zx and 2y for zy in equation (2),

rx×ry=2xi2yj+k

Modify equation (1) as follows.

SFdS=DF(rx×ry)dA

Substitute (x2i+xyj+zk) for F and (2xi2yj+k) for rx×ry ,

SFdS=D(x2i+xyj+zk)(2xi2yj+k)dA=D(2x32xy2+z)dA

Substitute x2+y2 for z ,

SFdS=D(2x32xy2+x2+y2)dA (4)

Find the value of limits as follows.

Consider the circle equation as follows.

x2+y2=z

Substitute 1 for z ,

x2+y2=1 (5)

Write the expression for circle equation.

x2+y2=r2 (6)

Compare equation (5) and (6).

r2=1r=1

Area of circle depends upon radius (r) and angle (θ) . The total angle required to complete one circle is 0 to 2π .

Substitute r2 for x2+y2 in equation (4),

SFdS=D(2x32xy2+r2)dA (7)

Write the expression for x and y in polar forms

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