Section 17.3 : Surface Integrals

1. 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
2. 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
3. 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
4. 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
5. 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
6. 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