This is a typical case of an incomplete reinstall. Frankly speaking, we often see that here.
If you use the Dell provided system restore media (of partition) to reinstall the original OS, you'll have sound. So it seems you didn't. Why you didn't is unclear.
If you use any other version of the OS or even another OS, it's your responsibility to install all drivers. With your service tag, you should be able to find them on www.dell.com.
I have a Dell Dimension E310. I reformatted and reloaded Windows XP. Now I have no sound. In device manager the IDT High Definition Audio CODEC has yellow exclamation point. I have tried to download drivers; it tells me I have the latest. I have tried to uninstall/reinstall; no luck. When I try troubleshoot it tells me device cannot start (code 10) The audio is integrated.