I have a Samsung Galaxy Y, and I have an application that has a scrollable activity. I wanted to take a screenshot but the "Home button - Power button" combination only captures the contents that are visible on the screen. What I wanted is to capture the full activity of the app including the unscrolled parts... I hope I explained it well ...
As said, a simple screenshot cannot do this for the mentioned reasons. What you could, however, do, is a ScreenCast -- which is easiest described as a video recording of your activities. AirDroid is one of the apps offering this for free (experimental, and root required). Searching the playstore for "screencast" will show you additional candidates. However, root will be required to operate them.
Aside from that, and not requiring root, is to make a series of screenshots, as suggested by t0mm13b already, and then using an image editor to connect them.
As was said - this cannot be done. I am just putting this in an answer form...
The reason being is that screenshots work by converting what is on the screen to an image - what is on the screen is pulled directly from the hardware.
Obviously, the screen doesn't know what is going to be displayed, a bit like we don't know exactly what is going to happen tomorrow (unless you're psychic, but we will ignore that for this analogy).
As such, there is no way to do this.
What you could do however, is to take many screenshots, scrolling down a bit, and then use a graphics package to stick the two screenshots together - may not look nice, but it will have to do I am afraid.