Should my iPhone 4 battery deplete this fast?

I just recently bought an iPhone 4 (new) and the battery goes in under 6 hours on a full charge! I’m not sure if this is normal, or if I have a faulty battery, or what. I usually get about 6 hours out of it while using mostly Wi-Fi and a little 3G, I text quite a bit and I will occasionally go onto Facebook. Also, I use Safari a bit to research my little dilemma. My screen is on about 50% brightness and not on Auto-brightness. When i only use 3G, I get about 4-5 hours. Does the battery take a while to reach its full potential, or should I be asking for a new phone?
I turned off Wi-Fi when using 3G but how do you know if apps are running in the background and how do you turn them off?

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