To make it simpler, the higher you go the harder you push?
NO, that's exactly the wrong way to do it.. the higher you go the more head voice you mix... I can elaborate on that but the harder you push the more strain you put on your chords and the more strained your chords are, the better the chance that they get messed up..
Why has nobody suggested lip rolls??? lip rolls are the EASIEST way to increase your range and one of the few exercises that you can really see/get results from in a reasonably short amount of time.
If anyone wants to learn to sing properly online, check out Brett Manning and his associates at SingingSuccess on youtube. I'm not trying to advertise his company or his products or anything, but he's given me some great tips over the years.
Quick Tip: If you want to increase your range so you can hit higher notes, try to sing as if you were speaking, as in keep your speaking tone, and as you go higher and higher, focus on ringing the sound from the back of your throat and letting some of that sound come out through your nose.
the "science" behind this: when a person speaks, his/her voice "resonates" through the nasal cavities and the throat. so when you talk, you should be able to feel vibrations on the bridge of your nose and on your adam's apple at the same time. Also, you should be able to feel vibrations on your head as well as on your chest. This mixture of resonance in your head and resonance in your chest is also known as "mixed voice" and that's the most commercially popular uhh 'technique' used. But it's not really a fancy technique because you use your "mixed voice" when you talk.
^Sorry if that didn't make sense. I tried my best to explain mixed voice in a nutshell but that whole thing might've been really confusing...