How to parameterize these surfaces?

[Sho Kano]

Homework Statement

Calculate $\iint { y+{ z }^{ 2 }ds }$ where the surface is the upper part of a hemisphere with radius a centered at the origin with $x\ge 0$

Homework Equations

Parameterizations:
$\sigma =\left< asin\phi cos\theta ,asin\phi sin\theta ,acos\phi \right> ,0\le \phi \le \frac { \pi }{ 2 } ,\frac { -\pi }{ 2 } \le \theta \le \frac { \pi }{ 2 } \\ N=(asin\phi )\sigma \\ \left| N \right| ={ a }^{ 2 }sin\phi \\ \\ \alpha =\left< rcos\theta ,rsin\theta ,0 \right> ,0\le r\le a,\frac { -\pi }{ 2 } \le \theta \le \frac { \pi }{ 2 } \\ N=-k\\ \left| N \right| =1$

The Attempt at a Solution

are these the right parameterizations?

 

[BvU]

Check for yourself: $\ \sigma =\left< a\sin\phi \cos\theta ,a\sin\phi \sin\theta ,a\cos\phi \right> \$seems right to me. For $\ \iint ds\$ you would then get $\ \pi a^2, \$ right ?

It is not clear to me what you do to express $\ ds \$. What is $N$ and what is the function of $N$ ?

 

[Sho Kano]

Check for yourself: $\ \sigma =\left< a\sin\phi \cos\theta ,a\sin\phi \sin\theta ,a\cos\phi \right> \$seems right to me. For $\ \iint ds\$ you would then get $\ \pi a^2, \$ right ?

It is not clear to me what you do to express $\ ds \$. What is $N$ and what is the function of $N$ ?

Oh sorry, by the integral I mean a surface integral. N is the normal. Both parameterizations seem right to me...i originally had $a$ instead of $r$ for the second parameterization. But that would just give me a circle, not a disk (a surface)

 

[BvU]

Are we mixing up two threads with almost the same title ?
Not clear to me why you need $N$ in this thread. But you sure need $ds$ and I haven't seen how you are going to express that in the parameters