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### Section 6-4 : Surface Integrals of Vector Fields

1. 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.
2. 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.
3. 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.
4. 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).
5. 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).
6. 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).
7. 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 $$S$$.
8. 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 $$S$$.
9. 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 $$S$$.