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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.

13. ∫∫s z2dS, S is the part of the paraboloid x = y2 + z2 given by 0 ≤ x ≤ 1

To determine

To find: The value of Sz2dS .

Explanation

Given data:

x=y2+z2 and 0x1 .

Formula used:

Sf(x,y,z)dS=Df(r(u,v))|ru×rv|dA (1)

ry=xyi+yyj+zyk (2)

rz=xzi+yzj+zzk (3)

Consider the general form of r(y,z) .

r(y,z)=xi+yj+zk

Substitute (y2+z2) for x ,

r(y,z)=(y2+z2)i+yj+zk

Find ry .

Substitute y2+z2 for x in equation (2),

ry=y(y2+z2)i+yyj+zyk=(2y+0)i+(1)j+(0)k=2yi+j

Find rz .

Substitute y2+z2 for x in equation (3),

rz=z(y2+z2)i+yzj+zzk=(0+2z)i+(0)j+(1)k=2zi+k

Find ru×rv .

ry×rz=(2yi+j)×(2zi+k)=|ijk2y102z01|=(10)i(2y0)j+(02z)k=i2yj2zk

Find |ry×rz| .

|ry×rz|=|i2yj2zk|=(1)2+(2y)2+(2z)2=1+4y2+4z2=1+4(y2+z2)

Find Sz2dS .

Modify equation (1) as follows.

Sz2dS=Dz2(|ry×rz|)dA

Apply limits and substitute 1+4(y2+z2) for |ry×rz| ,

Sz2dS=y2+z21z21+4(y2+z2)dA (4)

Let y=rcosθ , z=rsinθ and limits for r and θ are 0 to 1 and 0 to 2π .

Modify the equation (4) 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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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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