ยง16.5 Surfaces and Area

Lecture notes

Math3d is recommended for experimenting with implicit surfaces!

Key equations

\[\begin{aligned} A &= \iint_R |\r_u\times\r_v|\,du\,dv\\[1em] A &= \iint_R \frac{|\nabla F|}{|\nabla F\cdot\mathbf p|}\,dA\\[1em] A &= \iint_R \sqrt{1+f_x^2+f_y^2}\,dx\,dy \end{aligned}\]

Parametric vs implicit

Example: Find a parametrization of the cone \[ z = \sqrt{x^2 + y^2},\qquad 0\le z\le 1. \]

Surface area

The Grapher video gives an example of writing down the parametrization of a tangent plane. (This won't be on homework/exams though.)

Surface area of implicit surfaces

Examples

  1. Find the surface area of a cone of height $h$ and base radius $r.$
  2. Find the surface area of a sphere of radius $r.$
  3. Find a parametrization of a torus of major radius $R$ and minor radius $r$. Then compute its surface area.
  4. Find the surface area of the band cut from the paraboloid \[ z = x^2 + y^2 \] by the planes $z=1$ and $z=4$.