Question

Integrate ​G(x,y,z)equalsz over the parabolic cylinder yequalszsquared​, 0less than or equalsxless than or equals2​, 0less than or equalszless than or equalsStartFraction StartRoot 15 EndRoot Over 2 EndFraction . Write the integral ModifyingBelow Integral from nothing to nothing Integral from nothing to nothing With Upper S Upper G (x comma y comma z )d sigma as a double integral. ModifyingBelow Integral from nothing to nothing Integral from nothing to nothing With Upper S Upper G (x comma y comma z )d sigmaequalsIntegral from nothing to nothing Integral from nothing to nothing nothing dz dx.

Answer 1

I gather you're supposed to compute the integral of $G(x,y,z)=z$ over a surface $S$ that is the part of the parabolic cylinder $y=z^2$ with $0\le x\le2$ and $0\le z\le\frac{\sqrt{15}}2$.

We can parameterize $S$ by

$\vec s(x,z)=x\,\vec\imath+z^2\,\vec\jmath+z\,\vec k$

with the given constraints on $x$ and $z$. Take the normal vector to $S$ to be

$\vec s_x\times\vec s_z=-\vec\jmath+2z\,\vec k$

so that the surface element is

$\mathrm dS=\|\vec s_x\times\vec s_z\|\,\mathrm dx\,\mathrm dz=\sqrt{1+4z^2}\,\mathrm dx\,\mathrm dz$

Then in the integral, we have

$\displaystyle\iint_Sz\,\mathrm dS=\int_0^2\int_0^{\sqrt{15}/2}z\sqrt{1+4z^2}\,\mathrm dz\,\mathrm dx=\boxed{\frac{21}2}$