Evaluate \( \displaystyle \iint\limits_{S}{{2x - 3y + z\,dS}}\) where \(S\) is the portion of \(x + y + z = 2\) that is in the 1st octant.
Evaluate \( \displaystyle \iint\limits_{S}{{x + {y^2} + {z^2}\,dS}}\) where \(S\) is the portion of \(x = 4 - {y^2} - {z^2}\) that lies in front of \(x = - 2\).
Evaluate \( \displaystyle \iint\limits_{S}{{6\,dS}}\) where \(S\) is the portion of \(y = 4z + {x^3} + 6\) that lies over the region in the xz-plane with bounded by \(z = {x^3}\), \(x = 1\) and the \(x\)-axis.
Evaluate \( \displaystyle \iint\limits_{S}{{xyz\,dS}}\) where \(S\) is the portion of \({x^2} + {y^2} = 36\) between \(z = - 3\) and \(z = 1\).
Evaluate \( \displaystyle \iint\limits_{S}{{{z^2} + x\,dS}}\) where \(S\) is the portion of \({x^2} + {y^2} + {z^2} = 4\) with \(z \ge 0\).
Evaluate \( \displaystyle \iint\limits_{S}{{4y\,dS}}\) where \(S\) is the portion of \({x^2} + {z^2} = 9\) between \(y = 2\) and \(y = 10 - x\).
Evaluate \( \displaystyle \iint\limits_{S}{{z + 3\,dS}}\) where \(S\) is the surface of the solid bounded by \(z = 2{x^2} + 2{y^2} - 3\) and \(z = 1\). Note that both surfaces of this solid are included in \(S\).
Evaluate \( \displaystyle \iint\limits_{S}{{z\,dS}}\) where \(S\) is the surface of the solid bounded by \({y^2} + {z^2} = 4\), \(x = y - 3\), and \(x = 6 - z\). Note that all three surfaces of this solid are included in \(S\).
Evaluate \( \displaystyle \iint\limits_{S}{{4 + z\,dS}}\) where \(S\) is the portion of the sphere of radius 1 with \(z \ge 0\) and \(x \le 0\). Note that all three surfaces of this solid are included in \(S\).