try BBFlashBack ... it's not free either but yes it can actually record the whole screen and even a part of it ... just like the video that you want ...
Or with your iPad, there is a TubeDownloader app. I think that is what it is called. I don't know why Apple approved it, but you can download YouTube videos with it. Or, if you have it jailbroken, there are a few YouTube downloading apps like MxTube.
Unfortunately, my app TubeOrganizer does not have downloading capabilities, because Apple said they would reject such apps.