Evaluate the surface integral:
$$\iint _{ S }^{ }{ z } dS$$
Where S is the surface $x=y+3z^2$ for $0\le y\le 1$ and $0\le z\le 1$

Question

Evaluate the surface integral: 
$$\iint _{ S }^{  }{ z } dS$$
Where S is the surface $x=y+3z^2$ for $0\le y\le 1$ and $0\le z\le 1$

phoenixfire · Answer

I'm stuck. I think I have to use the formula $$\iint _{ S }^{  }{ f(x,y,z) } dS=\iint _{ D }^{  }{ f(x,y,g(x,y)) \sqrt { { (\frac { \delta z }{ \delta x } ) }^{ 2 }+{ (\frac { \delta z }{ \delta y } ) }^{ 2 }+1 }  } dA$$

But I keep getting lost. Can anyone help me out?

phoenixfire · Answer

Not sure if I should be able to project it onto the yz-plane and change the formula for x=g(y,z), etc. 
Or if I'm supposed to rearrange $x=y+3z^2$ for z, and continue with the normal way.

zarkon · Answer

$$\int_{ S }f(x,y,z)  dS=\iint_{ D } f(g(y,z),y,z) \sqrt {\left (\frac { \partial x }{ \partial y} \right)^{ 2 }+\left(\frac { \partial x }{ \partial z } \right)^{ 2}+1}~dA$$

phoenixfire · Answer

That's what I thought, but it didn't seem right. Oh well, I'll just go with it. Thanks @Zarkon

zarkon · Answer

you can always switch the roles of x and z and then use your original formula (same thing)