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### Section 6-3 : Surface Integrals

1. 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.
2. 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 $$x = - 2$$.
3. 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.
4. Evaluate $$\displaystyle \iint\limits_{S}{{xyz\,dS}}$$ where $$S$$ is the portion of $${x^2} + {y^2} = 36$$ between $$z = - 3$$ and $$z = 1$$.
5. Evaluate $$\displaystyle \iint\limits_{S}{{{z^2} + x\,dS}}$$ where $$S$$ is the portion of $${x^2} + {y^2} + {z^2} = 4$$ with $$z \ge 0$$.
6. Evaluate $$\displaystyle \iint\limits_{S}{{4y\,dS}}$$ where $$S$$ is the portion of $${x^2} + {z^2} = 9$$ between $$y = 2$$ and $$y = 10 - x$$.
7. 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 $$S$$.
8. 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 $$S$$.
9. 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 $$S$$.