   Chapter 16.7, Problem 6E

Chapter
Section
Textbook Problem

Evaluate the surface integral.6. ∬s xyz dS, S is the cone with parametric equations x = u cos v, y = u sin v, z = u, 0 ≤ u ≤ 1, 0 ≤ v ≤ π/2

To determine

To find: The value of SxyzdS .

Explanation

Given data:

x=ucosv , y=usinv , z=u , 0u1 and 0vπ2 .

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 ucosv for x , usinv for y and u for z in equation (2),

ru=u(ucosv)i+u(usinv)j+u(u)k=(cosv)u(u)i+(sinv)u(u)j+u(u)k=cosvi+sinvj+k

Find rv .

Substitute ucosv for x , usinv for y and u for z in equation (3),

rv=v(ucosv)i+v(usinv)j+v(u)k=(u)v(cosv)i+(u)v(sinv)j+(u)v(1)k=u(sinv)i+ucosvj+0k=usinvi+ucosvj

Find ru×rv .

ru×rv=(cosvi+sinvj+k)×(usinvi+ucosvj)=|ijkcosvsinv1usinvucosv0|=(0ucosv)i(0+usinv)j+(ucos2v+usin2v)k=ucosviusinvj+[u(cos2v+sin2v)]k

ru×rv=ucosviusinvj+uk {cos2θ+sin2θ=1}

Find |ru×rv|

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