Calculus III - Surface Integrals of Vector Fields (Assignment Problems)

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Section 17.4 : Surface Integrals of Vector Fields

Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = \left( {z - y} \right)\,\vec i + x\,\vec j + 4y\,\vec k$ and $S$ is the portion of $x + y + z = 2$ that is in the 1st octant oriented in the positive $z$ -axis direction.
Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = \left( {x - 4} \right)\,\vec i + z\,\vec j - y\,\vec k$ and $S$ is the portion of $x = 4 - {y^2} - {z^2}$ that lies in front of $x = - 2$ oriented in the negative $x$ -axis direction.
Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = \,\vec i + 4z\,\vec j + \left( {z - y} \right)\,\vec k$ and $S$ 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 oriented in the positive $y$ -axis direction.
Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = \left( {x + y} \right)\,\vec i + x\,\vec j + z{x^2}\,\vec k$ and $S$ is the portion of ${x^2} + {y^2} = 36$ between $z = - 3$ and $z = 1$ oriented outward (i.e. away from the $z$ -axis).
Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = z\,\vec i + 3\,\vec k$ and $S$ is the portion of ${x^2} + {y^2} + {z^2} = 4$ with $z \ge 0$ oriented outwards (i.e. away from the origin).
Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = - x\,\vec i + \left( {4 + y} \right)\,\vec j - z\,\vec k$ and $S$ is the portion of ${x^2} + {z^2} = 9$ between $y = 2$ and $y = 10 - x$ oriented inward (i.e. towards from the $y$ -axis).
Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = y\,\vec i + 2\,\vec j + {\left( {z + 3} \right)^2}\,\vec k$ and $S$ is the surface of the solid bounded by $z = 2{x^2} + 2{y^2} - 3$ and $z = 1$ with the negative orientation. Note that both surfaces of this solid are included in $S$ .
Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = \left( {x - y} \right)\,\vec i + z\,\vec j + y\,\vec k$ and $S$ is the surface of the solid bounded by ${y^2} + {z^2} = 4$ , $x = y - 3$ , and $x = 6 - z$ with the positive orientation. Note that all three surfaces of this solid are included in $S$ .
Evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}$ where $\vec F = y\,\vec i - 2\,\vec k$ and $S$ is the portion of the sphere of radius 1 with $z \ge 0$ and $x \le 0$ with the positive orientation. Note that all three surfaces of this solid are included in $S$ .