Mysql – Can MySQL use an index to calculate SUM

The following query is run in a MySQL database (all InnoDB tables, Percona 5.5), and is slow for some parameter values (i.e. for some large customers). It returns the total size of all files uploaded by a given customer:

SELECT SUM(document.size)
  FROM document
       INNER JOIN post ON post.id = document.id
       INNER JOIN page ON page.id = post.page_id
 WHERE page.customer_id=1234;

I tried denormalization — added document.customer_id column and index on document(customer_id, size) — and got much better query times. But I'd like to avoid denormalization if possible. I also tried adding just an index on document(size), but couldn't get MySQL to use it. The documentation states that every index in an InnoDB table also contains references to the primary key, so why did MySQL not use this new index when joining, and save itself the need to load document row data?

The schema looks like:

CREATE TABLE page (
  INT id NOT NULL,
  INT customer_id NOT NULL,
  ...
  PRIMARY KEY (id),
  INDEX ix_page_customer (customer_id)
  FOREIGN KEY (customer_id) REFERENCES customer (id)
)

CREATE TABLE post (
  INT id NOT NULL,
  ...
  PRIMARY KEY (id),
  INDEX ix_post_page (page_id),
  FOREIGN KEY (page_id) REFERENCES page (id)
)

CREATE TABLE document (
  INT id NOT NULL,
  INT size NOT NULL,
  ...
  PRIMARY KEY (id),
  FOREIGN KEY (id) REFERENCES post (id)
)

A customer contains many pages, a page contains many posts. Posts can be of different types (e.g. message, document). There are tables for each of these post subtypes, of which document is one. One of the larger customers has 435K posts (of which 8K are documents) spread over 25K pages.

The query is fairly fast for smaller customers (i.e lower pages*posts), but slow for larger customers. Here's EXPLAIN output for the query for a large customer:

+----+-------------+-----------+--------+--------------------------+------------------+---------+-----------------------+-------+-------------+
| id | select_type | table     | type   | possible_keys            | key              | key_len | ref                   | rows  | Extra       |
+----+-------------+-----------+--------+--------------------------+------------------+---------+-----------------------+-------+-------------+
|  1 | SIMPLE      | page      | ref    | PRIMARY,ix_page_customer | ix_page_customer | 4       | const                 | 51200 | Using index |
|  1 | SIMPLE      | post      | ref    | PRIMARY,ix_post_page     | ix_post_page     | 4       | database.page.page_id |     2 | Using index |
|  1 | SIMPLE      | document  | eq_ref | PRIMARY                  | PRIMARY          | 4       | database.post.post_id |     1 |             |
+----+-------------+-----------+--------+--------------------------+------------------+---------+-----------------------+-------+-------------+