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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.

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

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Chapter 16 Solutions

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