# How to parameterize these surfaces?

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1. Dec 13, 2016

### Sho Kano

1. The problem statement, all variables and given/known data
Calculate $\iint { y+{ z }^{ 2 }ds }$ where the surface is the upper part of a hemisphere with radius a centered at the origin with $x\ge 0$

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

3. The attempt at a solution
are these the right parameterizations?

2. Dec 14, 2016

### 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 $N$ ?

3. Dec 14, 2016

### Sho Kano

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)

4. Dec 14, 2016

### 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