   Chapter 16.7, Problem 18E

Chapter
Section
Textbook Problem

Evaluate the surface integral.18. ∫∫s (x + y + z) dS, S is the part of the half-cylinder x2 + z2 = 1. z ≥ 0, that lies between the planes y = 0 and y = 2

To determine

To find: The value of S(x+y+z)dS .

Explanation

Given data:

x2+z2=1 , z0 , y=0 and y=2 .

Formula used:

Sf(x,y,z)dS=Df(r(u,v))|ru×rv|dA (1)

ru=xui+yuj+zuk (2)

rv=xvi+yvj+zvk (3)

Let x=cosv , y=u , z=sinv and limits for u and v are 0 to 2 and 0 to 2π .

r(u,v)=cosvi+uj+sinvk

Find ru .

Substitute cosv for x , u for y and sinv for z in equation (2),

ru=u(cosv)i+u(u)j+u(sinv)k=(cosv)u(1)i+u(u)j+(sinv)u(1)k=j

Find rv .

Substitute cosv for x , u for y and sinv for z in equation (3),

rv=v(cosv)i+v(u)j+v(sinv)k=sinvi+(u)(0)j+cosvk=sinvi+cosvk

Find ru×rv .

ru×rv=(j)×(sinvi+cosvk)=|ijk010sinv0cosv|=(cosv0)i(00)j+(0+sinv)k=cosvi+sinvk

Find |ru×rv|

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