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

$\displaystyle z^2=x^2+y^2$

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 $\displaystyle z=v$

and $\displaystyle x=v\cos(t)$

and $\displaystyle y=v\sin(t)$

I chose this because

$\displaystyle x^2+y^2=v^2\cos^2(t)+v^2\sin^2(t)=v^2=z^2$

Is this correct?

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

$\displaystyle z=x^2+y^2$

I believe that what I would do is let

$\displaystyle z=v$

and

$\displaystyle x=\sqrt{v}\cos(t)$

and

$\displaystyle y=\sqrt{v}\sin(t)$

So we would get

$\displaystyle x^2+y^2=v\cos^2(t)+v\sin^2(t)=v=z$

But what if the quetion also stated that it is the portion of this curve below $\displaystyle z=4$?

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