Is swap area required? Can we install Ubuntu without a swap area?

No, you don't need a swap partition, as long as you never run out of RAM your system will work fine without it, but it can come in handy if you have less than 8GB of RAM and it is necessary for hibernation. For more information see this question: Do we still need swap partitions on desktop?

However, you can get around the 4 partition limit by creating an extended partition, and then creating a swap partition and a default partition (these "sub" partitions appear under the extended partition and are called "logical" partitions). This will be an option when you create your partitions in the installer. Check out this related question: My disk already has 4 primary partitions, how can I install Ubuntu?

You don't really need a swap partition. In our times computers have 4GB of RAM or more. That is normally enough for daily use.

But... If you do RAM heavy tasks your machine might run out of RAM... and crash.

In my opinion a good solution for you is to install Ubuntu on that single partition an add maybe 1GB of swap as a swapfile.

Here is a good toutorial for that... Ubuntu Linux Create and Add Swap File Tutorial

Another solution would be to enable zRam. Look here... How do I use zRam?

Swap partitions do two things: they can provide an overall speed boost by freeing up memory for more cache, and they can prevent rare disastrous out-of-memory situations where the system will start killing processes without warning.

If you have tons of RAM, you probably aren't needing it for a speed boost - it'll almost never be utilised. But it is still a good idea to have it for the other purpose: just in case you run out of memory completely. This can still happen even in these days of 8GB+ RAM, ie due to a rogue process, or a very memory-heavy process such as an image editor.

You don't need to use a partition. It is possible to have a swap file instead of a swap partition. Just don't set up any swap partition, and then set up a swap file later.