   Chapter 16.7, Problem 5E

Chapter
Section
Textbook Problem

Evaluate the surface integral.5. ∬s (x + y + z) dS,S is the parallelogram with parametric equations x = u + v, y = u − v, z = 1 + 2u + v, 0 ≤ u ≤ 2, 0 ≤ v ≤ 1

To determine

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

Explanation

Given data:

x=u+v , y=uv , z=1+2u+v , 0u2 and 0v1 .

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)

Find ru .

Substitute u+v for x , uv for y and 1+2u+v for z in equation (2),

ru=u(u+v)i+u(uv)j+u(1+2u+v)k=(1+0)i+(10)j+(0+2+0)k=i+j+2k

Find rv .

Substitute u+v for x , uv for y and 1+2u+v for z in equation (3),

rv=v(u+v)i+v(uv)j+v(1+2u+v)k=(0+1)i+(01)j+(0+0+1)k=ij+k

Find ru×rv .

ru×rv=(i+j+2k)×(ij+k)=|ijk112111|=(1+2)i(12)j+(11)k=3i+j2k

Find |ru×rv| .

|ru×rv|=|3i+j2k|=(3)2+(1)2+(2)2=9+1+4=14

Find S(x+y+z)dS

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