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Q. 34

Found in: Page 1119


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

fx,y,z=1x lnz+2y, where S is the surface given by ru,v=2ui+u+vj+2u-2vk for 3u7 and 2v10.

The required integral over the surface S is 24lnln(28)ln(12).

See the step by step solution

Step by Step Solution

Step 1. Given information.

Consider the given question,

fx,y,z=1x lnz+2y,ru,v=2ui+u+vj+2u-2vk

Step 2. Find the integral of fx,y,z over a smooth surface S.

If a surface S is parametrized by ru,v for u,vD, then the integral of fx,y,z over a smooth surface S is given below,

Sfx,y,zdS=fDxu,v,yu,v,zu,vru×rwdA ...... (i)

Now, find ru,rv,

role="math" localid="1650303986567" ru=uru,v=u2ui+u+vj+2u-2vk=2i+j+2krv=vru,v=v2ui+u+vj+2u-2vk=0i+j-2k

Step 3. Find ru×rv.

On solving ru×rv,



Consider the following function, localid="1650304882776" fx,y,z,

localid="1650304885354" fxu,v,yu,v,zu,v=12u ln2u-2v+2u+v=12u ln4u

Step 4. Consider the given function.

Substitute the values in equation (i),


Take the integral of f over the surface S,


Step 5. Continue solving the above equation.

On solving the above equation,


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