Example of Surface Integrals using the Transformation Approach

 

 

 

 

 

 

 

 

 

1.

 

Evaluate the surface integral                   IS   =  ∫  ∫  y2  dS

 

where   S  is the part of the sphere  x2  +  y2  +  z2  =  4   that lies inside the

cylinder  x2  +  y2  =  1  and above the xy-plane.

               

 

 

 

 

 

 

 

 

 

 

 

 

2.

 

 

 

Recall that calculation of surface area is a two step process.

 

Step 1:  Express f(x,y,z) in terms of the independent variables (in this case)  x and y.

 

So in this example      f(x,y,z)  =  y2

 

Step 2:  Write the surface area element,  dS  in terms of dA.   dS  =  | N | dA

In this example the projection of dS  is a circle of radius 1 in the xy-plane.

                          

 

Next calculate  N :         N(x,y)  =  rx  x  ry    where  r  =  x i + y j + z k

 

Click here to continue with this example.

 


Copyright © 2011 Richard C. Coddington

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