12.6 Videos Guide

12.6a

• Cylinder: A surface that consists of parallel lines through a plane curve

Exercises:

• Describe and sketch the surface $y=z^{2}$.

• Sketch a graph of the surface $z=\sin x$.

12.6b

• Equations of quadratic surfaces:

  • Ellipsoid: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$

  • Cone: $\frac{z^{2}}{c^{2}}=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$

  • Elliptic paraboloid: $\frac{z}{c}=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$

  • Hyperboloid of one sheet: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1$

  • Hyperbolic paraboloid: $\frac{z}{c}=\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}$

  • Hyperboloid of two sheets: $-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$

Exercises:

• Use traces to sketch and identify the surface $4x^{2}+9y^{2}+9z^{2}=36$.

12.6c

• Sketch the region bounded by $x^{2}+z^{2}=1$, $y+z=2$, and $y=0$.