Section 17.2 : 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 1^st 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 $x$-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