I don't think you need to learn the entire Extension 1 course, but here are the main topics I think you probably should familiarise with (in order):
- Further trigonometry (this links to many other topics)
- Integration techniques (this links to Applications of Calculus)
- Applications of Calculus (very important)
Those are the only three topics I'd recommend studying. Yes that's right, just three. However, here are a couple of more topics you may benefit from looking at if you have time:
- Permutations and Combinations (links to Binomial Theorem)
- Binomial Theorem (links to Further Probability)
- Further Probability (not that difficult of a topic itself, but its prerequisite topics may be somewhat challenging)
So there you go. The latter three aren't too relevant, so you'll be probably be fine if you don't do them (although one can never quite get away from statistical and probabilistic analysis... even in an engineering degree).
On a side note, if you are learning by yourself, I strongly recommend the new Fitzpatrick textbook. There are quite a few students selling their textbooks for less than half the retail price for a new one. The questions in the exercises of the Fitzpatrick textbook are highly relevant and their explanations are better than many of the other textbooks I've seen (plus it looks and feels great!).