15.3 Surface Integrals: Surface Area and Flux


1. Evaluate where and is that part of the plane which is  located in the first octant.

 Sol) .


2. If is the entire surface of the cube bounded by , , , , , and and then evaluate .

 Sol)

                                       
 
 
 
 
 
  Therefore .


3. Let be the surface of the cylinder included in the first octant between and . Evaluate where .

 Sol)

   


 

 ,

 ,

 ,

 

  Therefore .


4. Find the surface integral over the parallelepiped , , , , , when .

 Sol)

,

     ,

     ,

     ,

     ,

     .

     Therefore .


5. If is the surface of the sphere and , evaluate .

 

 

 


6. Evaluate .

   Here and is the boundary of the solid surrounded by and .

 Sol)

 1) :

        .


 2) : .

 3) : .