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### Section 6-2 : Parametric Surfaces

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

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