Total of orders open on a given date for each date in a date range
If your priority is speed of selects, the following approach allows for extremely fast selects. Instead of storing a period, you want to store two events (period start and period end). Change column is 1 when the period begins, and is -1 when the period ends. If more than one event occurs on the same day, they must have different EventNumberPerDay. RunningTotal is the number of open peroids after the event has happened:

CREATE TABLE dbo.Events(
 PeriodId INT NOT NULL,
 Change SMALLINT NOT NULL,
 ChangeDate DATE NOT NULL,
 EventNumberPerDay INT NOT NULL,
 RunningTotal INT NOT NULL);
GO

INSERT dbo.Events
        ( PeriodId ,
          Change ,
          ChangeDate ,
          EventNumberPerDay,
          RunningTotal
        )
-- first period begins    
VALUES  ( 1 , 1, '20120801', 1, 1),
-- second period begins on the same day
(2, 1, '20120801', 2, 2),
-- third period begins
(3,1,'20120803',1, 3),
-- second period ends on the same day
(2,-1,'20120803',2, 2),
-- fourth period begins
(4,1,'20120804',1,3),
-- fourth period ends
(4,-1,'20120805',1,2),
-- first period ends
(1, -1, '20120808',1,  1),
-- third period ends
(3, -1, '20120809',1,  0);
GO


Also you need a calendar table:

CREATE TABLE dbo.Calendar([Date] DATE NOT NULL);
GO

INSERT INTO dbo.Calendar([Date])
VALUES('20120801'),
('20120802'),
('20120803'),
('20120804'),
('20120805'),
('20120806'),
('20120807'),
('20120808'),
('20120809'),
('20120810'),
('20120811');


Once that is accomplished, then your select is very simple and very fast:

SELECT  [Date] ,
        coalesce(RunningTotal, 0) AS NumOpenIntervals
FROM    dbo.Calendar
        OUTER APPLY ( SELECT TOP ( 1 )
                                RunningTotal
                      FROM      dbo.Events
                      WHERE     ChangeDate &lt;= [Date]
                      ORDER BY  ChangeDate DESC, EventNumberPerDay DESC
                    ) AS t
ORDER BY [Date]


Of course, this query is only correct if the data in Events table is valid. We can use constraints to ensure 100% data integrity. I can explain how if you are interested.

Another alternative is to just load your raw data, your periods, into a client application - your problem is absolutely trivial in C++/C#/Java.

Yet another approach is to use an RDBMS with fast cursors such as Oracle - that will allow you to just write a simple cursor and enjoy good performance, but still not always as good as my first solution.