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

1. PhoenixFire

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?

2. PhoenixFire

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.

3. Zarkon

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

4. PhoenixFire

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

5. Zarkon

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