Area

This problem was taken from the Collegiate Challenge Finals.

Class name: Area
Method name: totalEnclosed
Parameters: int, int, String 
Returns: int

Implement a class Area, which contains a method totalEnclosed.  totalEnclosed
returns the sum of the union of areas enclosed by a path on the coordinate
plane.  The path (call it the "perimeter") is composed of horizontal and
vertical lines connecting ordered integer pairs. The perimeter is defined by a
starting point (x,y) (x and y are integers) and a series of commands.  The
commands are U, D, R, and L.  U moves 1 unit up.  D moves 1 unit down.  R moves
1 unit right.  L moves 1 unit left.  The method takes as parameters two ints
that are the starting (x,y) coordinates and a String that is the string of 1
character commands.  The method starts at the starting point and successively
processes the commands in the String from left to right and returns the area of
the union of the shapes enclosed by the resulting perimeter.  

*Enclosed areas must lie within the domain -10 <= x <= 10 and -10 <= y <= 10.
If any point on the perimeter is beyond this domain, the method concludes there
is no enclosed area and returns 0.
*A point is in an enclosed area if and only if every continuous path from the
point to outside the domain (-10 <= x <= 10 and -10 <= y <= 10) crosses the
perimeter.

Another way to look at this is: return the number of unit squares in the domain
that are enclosed by the perimeter.

The method signature is (be sure your method is public):
int totalEnclosed(int startX, int startY, String commands)

TopCoder will ensure the following:
*startX must satisfy -10 <= startX <= 10.
*startY must satisfy -10 <= startY <= 10.
*commands contains only the characters U, D, L, and R and has length less than
1000 characters.

Note:
-The solution must run in under 6 seconds on TopCoder's tester.

Examples:
-If the starting point is (0,0) and the commands are RRUULLDDDDL, the enclosed
area is a square from (0,0) to (2,2). The area of this square is 4, and the
method should return 4.

-If the starting point is (3,3) and the commands are LDRULLLDDRUU, there are
two enclosed areas, a square from (2,2) to (3,3) and a rectangle from (0,1) to
(1,3). The areas are 1 and 2, and the method should return 1+2=3.

-If the starting point is (0,0) and the commands are LLUURRDDLURD, the enclosed
area is just the square from (-2,0) to (0,2) and the method should return the
square's area, 4.  The area from (-1,0) to (0,1) is also enclosed by another
path, but is not counted twice because the method must return the area of the
union.

-If the starting point is (10,10) and the commands are URDL, the path goes out
of the domain and the method should return 0.