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### Section 6-3 : 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