The task at hand is as follows: use the iPad as you would a whiteboard in a classroom and record your writing and your voice while you are doing it. Here's a 20-second example of this task accomplished with Explain Everything app:
In the PC/Mac/Online world, screencasting is a recording of anything you do on your computer screen. I have not seen a true screencasting app for the iPad. The existing apps restrict the recording to user actions within the screencasting app itself. In contrast, on a computer, a screencasting app can record a user interacting with any computer program, such as a PowerPoint presentation, a movie on the Web, a Microsoft Word document, etc. A post on creating screencasts online appears here.
The best of the available "screencasting" apps for the iPad is Explain Everything ($3). Here's the app's description on iTunes:
"Explain Everything is an easy-to-use design tool that lets you annotate, animate, and narrate explanations and presentations. You can create dynamic interactive lessons, activities, assessments, and tutorials using Explain Everything's flexible and integrated design. Use Explain Everything as an interactive whiteboard using the iPad2 video display.
Explain Everything records on-screen drawing, annotation, object movement and captures audio via the iPad microphone. Import Photos, PDF, PPT, XLS, RTF from Dropbox, Evernote, Email, iPad photo roll and iPad 2 camera. Export MP4 movie files, PNG image files, PDFs and share the .XPL project file with others for collaboration."
You can start with a blank whiteboard, a photo, a Powerpoint presentation, a PDF file, etc. You can then talk while doing any of the following: draw freehand, draw shapes, type, zoom in/out, edit/erase/rearrange objects, insert photos. At the end, you can export the screencast as a movie to upload to YouTube, your blog, your website, etc. It is well-integrated with Dropbox, both for import and for export, as well as many other popular services.
If you need a free app, Screenchomp, Showme, and Educreations are very similar to each other and very good. The limitation with all three is that you can only write freehand but not type.