Difference between 'part' and 'a part'?

This question may seem to be very simple, but something I get confused whenever I want to speak.

I read a book entitled "re-start your English", and saw a sentence.

This is a leg. It is part of your body.

This is a neck. It is 'a' part of your body.

Why didn't they use 'a' in the first sentence? And why did they use one in the second one?

I'm not a native, so I hope you will please understand, even if there's something awkward about my question.

Solution 1:

When part is used without the article it has the meaning of section or segment - it is less than the whole object.

A part has the meaning of a specific sub unit. It normally isn't used for something which isn't an identifiable thing all by itself (though just to be confusing sometimes it is.)

Solution 2:

Generally, you can use either.

'My arm is part of my body' and 'My arm is a part of my body' are both everyday uses to native English speakers. They mean more or less the same thing. There is a slightly nuanced difference in meaning but one which even a very experienced speaker would find it difficult to explain. So I wouldn't worry about that.

I presume in your first example it is meant to read 'This is a leg. It is part of your body'.

Do remember always to start a sentence with a capital letter. You will see that I have extensively edited the OP, but I didn't interfere with the sentence examples you gave.