By getting all of the extra work done, you risk alienating your co-worker. You might be called a show-off. Not knowing how your co-worker would handle the work might leave you getting the work done, but not how she would have done it. If your co-worker has been doing a good job, this might actually hurt you in comparison.

By getting most of the extra work done, you show that you can prioritize tasks.

By getting all of your work done and some of the extra work done, you show that you can do your work and more.

If you were your boss, what you like to hear?

Some people might feel overwhelmed and say, "I'll get done what I can." Some people might resent the extra work, and say, "I've got a bunch of deadlines. I just don't know what I can do." Some people might respond with enthusiasm, "Wow, a challenge. It sounds like fun. I'll see what I can do."

There is a business training video that helps employees develop into the kind of workers that managers like to see. It's called Tough Times: Making the Most of Your Job.

"Tough Times: Making the Most of Your Job discusses communication skills, flexibility, company knowledge, and the willingness to take on extra responsibilities and learn new skills. These are stressed as stepping stones to career advancement and job security."
-- Ad copy for Tough Times: Making the Most of Your Job

Tough Times gives some great information for those who want to move up within their own organization.

About the extra work question, I think I would take a quick look at what the extra work entails and prioritize the items along with my own jobs. There might be some tasks that require more information, which only the co-worker might supply. These I would put on the bottom of the pile to get to if time permits. There might be some tasks that combine existing jobs on your schedule and the extra work.

At the end of the day, I would summarize in a memo what was done. I'd keep a file copy, and give one to the boss and one to the co-worker. It would be good to mention how I accomplished what I did. If I put aside some of my own busy work that could easily wait for another day, I'd write it down. If I accomplished some of the extra work, but had the benefit of my co-worker's notes or remembered conversations that should be noted as well. And finally, I think I would add a proviso that although I may have completed the extra work, it would probably have been done better by my co-worker.

I think this approach covers you, your boss, and your co-worker. It shows concern for the feelings of your co-worker, and your desire to perform. It's also a grand example of initiative, reliability, planning, decisiveness, communication and teamwork.

For someone who wants to get ahead this type of opportunity exists almost everyday. You don't need to wait for a co-worker to be sick. You can collaborate with a co-worker to get more things done, you can ask your boss for more work to show what you can do, you can find your own tasks to perform that will help the organization. Extra work is always there. Seek it and succeed.