Another service from Omega

Two Cylinders to Worry About


*****

Problem

Find the surface area of the portion of the cylinder

> y^2+z^2 = 9;

                                   2    2
                                  y  + z  = 9

that lies inside the cylinder

> x^2+z^2 = 9;

                                   2    2
                                  x  + z  = 9


Solution


The part of the surface S above the xy plane is characterized by
z = sqrt(9-y^2) with -3 < y < 3 and -|y| < x < |y|

> with(linalg): i := vector([1,0,0]): j:=vector([0,1,0]): k:=vector([0,0,1]):
> r = x*i + y*j + sqrt(9-y^2)*k;

                                               2 1/2
                         r = x i + y j + (9 - y )    k

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

> z := sqrt(9-y^2): dS := sqrt(1+diff(z,x)^2+diff(z,y)^2);

                                   /       2  \1/2
                                   |      y   |
                             dS := |1 + ------|
                                   |         2|
                                   \    9 - y /

the total area is then,

> Answer := 2*Int(Int(dS,x=-abs(y)..abs(y)),y=-3..3) =
> 2*int(int(dS,x=-abs(y)..abs(y)),y=-3..3);

 Answer :=

         3    abs(y)                             3
         /      /    /       2  \1/2             /
        |      |     |      y   |               |    /      1   \1/2
     2  |      |     |1 + ------|    dx dy = 2  |  6 |- --------|    abs(y) dy
        |      |     |         2|               |    |         2|
       /      /      \    9 - y /              /     \  - 9 + y /
       -3  - abs(y)                            -3

and this evaluates to:

> evalf(rhs("));

                                  72.00000000

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Wed Nov 13 10:08:30 EST 1996