Evaluate \( \displaystyle \iint\limits_{S}{{z + 3y - {x^2}\,dS}}\) where \(S\) is the portion of \(z = 2 - 3y + {x^2}\) that lies over the triangle in the \(xy\)-plane with vertices \(\left( {0,0} \right)\), \(\left( {2,0} \right)\) and \(\left( {2, - 4} \right)\). Solution
Evaluate \( \displaystyle \iint\limits_{S}{{40y\,dS}}\) where \(S\) is the portion of \(y = 3{x^2} + 3{z^2}\) that lies behind \(y = 6\). Solution
Evaluate \( \displaystyle \iint\limits_{S}{{2y\,dS}}\) where \(S\) is the portion of \({y^2} + {z^2} = 4\) between \(x = 0\) and \(x = 3 - z\). Solution
Evaluate \( \displaystyle \iint\limits_{S}{{xz\,dS}}\) where \(S\) is the portion of the sphere of radius 3 with \(x \le 0\), \(y \ge 0\) and \(z \ge 0\). Solution
Evaluate \( \displaystyle \iint\limits_{S}{{yz + 4xy\,dS}}\) where \(S\) is the surface of the solid bounded by \(4x + 2y + z = 8\), \(z = 0\), \(y = 0\) and \(x = 0\). Note that all four surfaces of this solid are included in \(S\). Solution
Evaluate \( \displaystyle \iint\limits_{S}{{x - z\,dS}}\) where \(S\) is the surface of the solid bounded by \({x^2} + {y^2} = 4\), \(z = x - 3\), and \(z = x + 2\). Note that all three surfaces of this solid are included in \(S\). Solution