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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Evaluate the surface integral.

27.S z dS, where S is the part of the paraboloid z = x2 + y2 that lies under the plane z = 4

To determine

To Evaluate: The surface integral of SzdS .

Explanation

Given data:

z=x2+y2 , z=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(x2)=2xddy(x2)=0ddy(y2)=2yddx(y2)=0

Consider the expression as follows.

z=x2+y2 (4)

Differentiate equation (4) with respect to x .

zx=2x

Differentiate equation (4) with respect to y .

zy=2y

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

rx×ry=2xi2yj+k

Substitute 2x for zx and 2y for zy in equation (3),

|rx×ry|=(2x)2+(2y)2+1=4x2+4y2+1

Modify equation (1) as follows.

SzdS=Dz|rx×ry|dA

Substitute x2+y2 for z and 4x2+4y2+1 for |rx×ry| ,

SzdS=D(x2+y2)(4x2+4y2+1)dA

SzdS=D(x2+y2)(4(x2+y2)+1)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 .

Substitute r2 for x2+y2 in equation (5),

SzdS=Dr2(4r2+1)dA

Apply limits and modify equation as follows.

SzdS=02π02r2(4r2+1)rdrdθ=02π02r3(4r2+1)drdθ=02r3(4r2+1)dr[θ]02π=02r3(4r2+1)dr[2π0]

Simplify equation as follows.

SzdS SzdS=2π02r3(4r2+1)dr

SzdS=2π02r2(4r2+1)rdr (8)

Consider the expression for u as follows

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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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