Query to count ACTIVE records for each month

Question

All,
I have a requirement to count the number of active customers for each month of an year. Let's say, we have 3 fields in the table:
ID               ActiveStart             ActiveEnd
1                 1/1/2014               11/30/2014
In this scenario, the member with ID was active for all the months (Jan-November) EXCEPT, December. So, the result set I am expecting for the Month of January to December is like below:
Month       CountOfMembers
Jan            1
Feb           1
Mar           1
April          1
May           1
June          1
July           1
August      1
Sept          1
Oct            1
Nov           1
Dec           0 (Since, the member was NOT active in this month)
Similarly, if you have another member whose start date is 1/1/2014 and termination is '5/31/2014', then, I would need to add 1 for months of January till May, the member WON'T be counted for the rest of the months. So, in this case, the count will be come
"2" for months January till May, it will remain "1" for June till November and still "O" for December.

Answer 1

>>I have a requirement to count the number of active customers for each month of an year. Let's say, we have 3 fields [sic: columns are not fields] in the table [sic: tables have names]: <<
Please follow basic Netiquette and post the DDL we need to answer this. Follow industry and ANSI-ISO standards in your data. You should follow ISO-11179 rules for naming data elements. You should follow ISO-8601 rules for displaying temporal data. We need to
know the data types, keys and constraints on the table. Avoid dialect in favor of ANSI-ISO Standard SQL. And you need to read and download the PDF for: 
https:--www.simple-talk.com-books-sql-books-119-sql-code-smells-
There is no magical universal “id” in RDBMS, tables have names and columns have constraints and types. Most of the work in SQL is done in the DDL. Here is a possible schema. 
CREATE TABLE Duty_Roster
(member_nbr INTEGER NOT NULL PRIMARY KEY,
 duty_start_date DATE NOT NULL,
 duty_end_date DATE NOT NULL,
CHECK (duty_start_date <= duty_end_date),
INSERT INTO Duty_Roster
VALUES 
(1, '2014-01-01', '2014-11-30'),
(2, '2014-01-01', '2014-05-31');              
>> In this scenario, the member with ID was active for all the months (Jan-November) EXCEPT, December. So, the result set I am expecting for the Month of January to December is like below: <<
A month is a temporal measurement that makes no sense without a year. But if you knew the ISO temporal model, you would see that you have what is called an INTERVAL data type. T-SQL does not support it yet. 
Since SQL is a database language, we prefer to do look ups and not calculations. They can be optimized while temporal math messes up optimization. A useful idiom is a report period calendar that everyone uses so there is no way to get disagreements in the DML.
The report period table gives a name to a range of dates that is common to the entire enterprise. 
CREATE TABLE Month_Periods
(month_name CHAR(10) NOT NULL PRIMARY KEY
   CHECK (month_name LIKE  '[12][0-9][0-9][0-9]-[01][0-9]-00'),
 month_start_date DATE NOT NULL,
 month_end_date DATE NOT NULL,
  CONSTRAINT date_ordering
    CHECK (month_start_date <= month_end_date),
etc);
These report periods can overlap or have gaps. I like the MySQL convention of using double zeroes for months and years, That is 'yyyy-mm-00' for a month within a year and 'yyyy-00-00' for the whole year. The advantages are that it will sort with the ISO-8601
data format required by Standard SQL and it is language independent. The pattern for validation is '[12][0-9][0-9][0-9]-00-00' and '[12][0-9][0-9][0-9]-[01][0-9]-00'
SELECT M.month_name, COUNT(R.member_nbr) AS member_cnt
  FROM Month_Periods AS M,
       Duty_Roster AS R
 WHERE M.month_start_date 
       BETWEEN R.duty_start_date AND R.duty_end_date
   AND M.month_end_date 
       BETWEEN R.duty_start_date AND R.duty_end_date;
We had no business rules about partial months, so I guessed. 
--CELKO-- Books in Celko Series for Morgan-Kaufmann Publishing: Analytics and OLAP in SQL / Data and Databases: Concepts in Practice Data / Measurements and Standards in SQL SQL for Smarties / SQL Programming Style / SQL Puzzles and Answers / Thinking
in Sets / Trees and Hierarchies in SQL