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Section 17.2 : 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 1^st 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 1^st 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 \({u^2} + {v^2} = 16\). \[\vec r\left( {u,v} \right) = \left\langle {uv,1 - 3v,2 + 3u} \right\rangle \]