Another service from Omega

Computing Surface Areas


*****


Problem1


What is the surface area of the portion of the surface

> z := x^2-y^2;

                                        2    2
                                  z := x  - y

that lies inside the cylinder,

> x^2+y^2=9;

                                   2    2
                                  x  + y  = 9


Solution:


The element of surface area (sans dxdy) is given by,

> dS := sqrt(1+diff(z,x)^2+diff(z,y)^2);

                                         2      2 1/2
                           dS := (1 + 4 x  + 4 y )
> Answer := Int(Int(dS,y=-sqrt(9-x^2)..sqrt(9-x^2)),x=-3..3) =
> evalf( int(int(dS,y=-sqrt(9-x^2)..sqrt(9-x^2)),x=-3..3));
                            2 1/2
                 3    (9 - x )
                 /         /
                |         |               2      2 1/2
     Answer :=  |         |       (1 + 4 x  + 4 y )    dy dx = 117.3187007
                |         |
               /         /
               -3          2 1/2
                   - (9 - x )

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Wed Nov 13 15:59:34 EST 1996