   Chapter 16, Problem 28RE

Chapter
Section
Textbook Problem

Evaluate the surface integral.28. ∬s (x2z + y2z)dS, where S is the part of the plane z = 4 + x + y that lies inside the cylinder x2 + y2 = 4

To determine

To Evaluate: The surface integral of S(x2z+y2z)dS .

Explanation

Given data:

z=4+x+y , x2+y2=4

Formula used:

Write the expression for surface integral.

Sf(x,y,z)dS=Df(r(u,v))|ru×rv|dA (1)

Write the expression for rx×ry .

rx×ry=zxizyj+k (2)

Write the expression for |rx×ry| .

|rx×ry|=(zx)2+(zy)2+1 (3)

Write the required differentiation formulae.

ddx(x)=1ddy(x)=0ddy(y)=1ddx(y)=0

Consider the expression as follows.

z=4+x+y (4)

Differentiate equation (4) with respect to x .

zx=1

Differentiate equation (4) with respect to y .

zy=1

Substitute 1 for zx and 1 for zy in equation (2),

rx×ry=ij+k

Substitute 1 for zx and 1 for zy in equation (3),

|rx×ry|=(1)2+(1)2+1=3

Modify equation (1) as follows.

S(x2z+y2z)dS=D(x2+y2)z|rx×ry|dA

Substitute 4+x+y for z and 3 for |rx×ry| ,

S(x2z+y2z)dS=D(x2+y2)(4+x+y)(3)dA (5)

Find the value of limits as follows.

Consider the circle equation as follows.

x2+y2=4 (6)

Write the expression for circle equation.

x2+y2=r2 (7)

Compare equation (6) and (7).

r2=4r=2

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

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

Find f g h. 42. f(x) = tan x, g(x)=xx1,h(x)=x3

Single Variable Calculus: Early Transcendentals, Volume I

Finding Intercepts In Exercises 19-28, find any intercepts. x2yx2+4y=0

Calculus: Early Transcendental Functions (MindTap Course List)

The range of is: (−∞,∞) [0, ∞) (0, ∞) [1, ∞]

Study Guide for Stewart's Multivariable Calculus, 8th

True or False: If f(x) = f (−x) for all x then .

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