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Section 6-2 : Parametric Surfaces

For problems 1 – 6 write down a set of parametric equations for the given surface.

  1. The plane \(7x + 3y + 4z = 15\). Solution
  2. The portion of the plane \(7x + 3y + 4z = 15\) that lies in the 1st octant. Solution
  3. The cylinder \({x^2} + {y^2} = 5\) for \( - 1 \le z \le 6\). Solution
  4. The portion of \(y = 4 - {x^2} - {z^2}\) that is in front of \(y = - 6\). Solution
  5. The portion of the sphere of radius 6 with \(x \ge 0\). Solution
  6. The tangent plane to the surface given by the following parametric equation at the point \(\left( {8,14,2} \right)\). \[\vec r\left( {u,v} \right) = \left( {{u^2} + 2u} \right)\vec i + \left( {3v - 2u} \right)\vec j + \left( {6v - 10} \right)\vec k\] Solution
  7. Determine the surface area of the portion of \(2x + 3y + 6z = 9\) that is inside the cylinder \({x^2} + {y^2} = 7\). Solution
  8. Determine the surface area of the portion of \({x^2} + {y^2} + {z^2} = 25\) with \(z \le 0\). Solution
  9. Determine the surface area of the portion of \(z = 3 + 2y + \frac{1}{4}{x^4}\) that is above the region in the \(xy\)-plane bounded by \(y = {x^5}\), \(x = 1\) and the \(y\)-axis. Solution
  10. Determine the surface area of the portion of the surface given by the following parametric equation that lies inside the cylinder \({u^2} + {v^2} = 4\). \[\vec r\left( {u,v} \right) = \left\langle {2u,vu,1 - 2v} \right\rangle \] Solution