My book skipped this section, but I think it is important, and I am having difficulty finding resources online. So if this is incorrect bare with me.

Say we have a curve defined by

Now we want to parametrize this, but we do not want z's parameter to be dependent on x and y's. So is this how you would paremetrize it?

Let

and

and

I chose this because

Is this correct?

And secondly how do we integrate restrictions into this, for example

I believe that what I would do is let

and

and

So we would get

But what if the quetion also stated that it is the portion of this curve below ?

If someone could tell me if I am doing this right, and then maybe give me a hint for how to do the last part, that would be great.

Thanks in advance,

Mathstud