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### Section 6-2 : Parametric Surfaces

For problems 1 – 10 write down a set of parametric equations for the given surface.

1. The plane containing the three points $$\left( {1,4, - 2} \right)$$, $$\left( { - 3,0,1} \right)$$ and $$\left( {2,4, - 5} \right)$$.
2. The portion of the plane $$x + 9y + 3z = 8$$ that lies in the 1st octant.
3. The portion of $$x = 2{y^2} + 2{z^2} - 7$$ that is behind $$x = 5$$.
4. The portion of $$y = 10 - 3{x^2} - 3{z^2}$$ that is in front of the $$xz$$-plane.
5. The cylinder $${x^2} + {z^2} = 121$$.
6. The cylinder $${y^2} + {z^2} = 6$$ for $$2 \le x \le 9$$.
7. The sphere $${x^2} + {y^2} + {z^2} = 17$$.
8. The portion of the sphere of radius 3 with $$y \ge 0$$ and $$z \le 0$$.
9. The tangent plane to the surface given by the following parametric equation at the point $$\left( { - 5,4, - 12} \right)$$. $\vec r\left( {u,v} \right) = \left( {u + 2v} \right)\vec i + \left( {{u^2} + 3} \right)\vec j - 3{v^2}\vec k$
10. The tangent plane to the surface given by the following parametric equation at the point $$\left( {1, - 11,19} \right)$$. $\vec r\left( {u,v} \right) = \left\langle {{{\bf{e}}^{6 - 2v}},{u^2} - 15,1 - u{v^2}} \right\rangle$
11. Determine the surface area of the portion of $$3x + 3y + 4z = 16$$ that is in the 1st octant.
12. Determine the surface area of the portion of $$x + 4y + 8z = 4$$ that is inside the cylinder $${x^2} + {y^2} = 16$$.
13. Determine the surface area of the portion of $$z = 6y + 2{x^2}$$ that is above the triangle in the $$xy$$-plane with vertices $$\left( {0,0} \right)$$, $$\left( {8,0} \right)$$ and $$\left( {8,2} \right)$$.
14. Determine the surface area of the portion of $$x = 6 - {y^2} - {z^2}$$ that is in front of $$x = 2$$ with $$y \ge 0$$.
15. Determine the surface area of the portion of $${x^2} + {y^2} + {z^2} = 11$$ with $$x \ge 0$$, $$y \le 0$$ and $$z \ge 0$$.
16. Determine the surface area of the portion of the surface given by the following parametric equation that lies above the triangle in the $$uv$$-plane with vertices$$\left( {0,0} \right)$$, $$\left( {10,2} \right)$$ and $$\left( {0,2} \right)$$. $\vec r\left( {u,v} \right) = \left\langle {{v^2},3v,2u} \right\rangle$
17. Determine the surface area of the portion of the surface given by the following parametric equation that lies above the region in the $$uv$$-plane bounded by $$\displaystyle v = \frac{3}{2}{u^2}$$, $$u = 2$$ and the $$u$$-axis. $\vec r\left( {u,v} \right) = \left\langle {uv,3uv,v} \right\rangle$
18. Determine the surface area of the portion of the surface given by the following parametric equation that lies inside the cylinder $${u^2} + {v^2} = 16$$. $\vec r\left( {u,v} \right) = \left\langle {uv,1 - 3v,2 + 3u} \right\rangle$