How to check what causes iPhone battery to drain out?
There is a general tips page on maintaining great battery life, as well as general terms about battery replacement.

Personally: This sounds like a bad battery. It sounds like it just plainly is not holding a charge after a certain discharge point. Remember that iOS multitasking is not true multitasking. Barring streaming audio (actually persistent audio in general), there is (almost) NOTHING that your phone can constantly do for more than 20 minutes at a time.

I've streamed audio for hours (hours being... maybe 5?) during the day at ~40% and it did get under 20% by the time I was done, this was both on an iPad, and roughly equivalent on an iPhone 4. Connected WiFi, 3G (on the iPhone) enabled but not being the active radio in use, of course.

I don't know that you're able to see what's draining the battery, Apple forbids "task management apps". A possible workaround to see if it is app involved or a bad battery, is to drain your phone to a similar level, open the fast app switcher bar (double tap the home button), tap-hold on any application in the task bar until they begin to "jiggle", then press the white minus inside the red circle on EVERYTHING. This will kill any and all tasks that aren't first-party. (For example, it will not stop the occasional housekeeping/tasks that Mail, Phone, Safari, or other built-in apps are allowed to do.)

Having said that, to properly test, ensure that mail's auto-fetch is off (or in push only mode at best), make sure that you have no tabs in Safari with "live" or automatically refreshing content. Safari IS ALLOWED TO BACKGROUND, and has always been able to, to the best of my knowledge. At least since iOS 2.something (where I started with an iPhone 3G).