SUNY Geneseo Department of Mathematics

Surface Area

Friday, December 9

Math 223 01
Fall 2022
Prof. Doug Baldwin

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Surface Area

Based on “Surface Area of a Parametric Surface” in section 5.6 of the textbook.

Key Ideas

The definition of a parametric surface (see Wednesday’s notes).

The equation for surface area, which builds on lots of previous vector operations, vector calculus, etc: for surface r(u,v), on parameter domain D…

Surface area is double integral over D of magnitude of R sub U cross R sub V

Examples of applications to various shapes.

Examples

Parameterize the surface (without ends) of a cylinder of radius r and height h, and use the integral formula for surface area to derive the equation S = 2πrh for that cylinder’s surface area (textbook exercise 5.6.6).

The first step is to come up with a parametric description of the cylinder. Think of it as a parametric circle, with a z coordinate that lets it extend over a third dimension:

Parametric cylinder is vector R cosine Theta comma R sine Theta comma Z

Then we needed the derivatives of the surface with respect to each variable:

Derivative of S with respect to Theta is negative R sine Theta, R cosine Theta, 0; with respect to Z is 0, 0, 1

From the derivatives, we worked out their cross product:

Cross product of derivatives of S with respect to Theta and Z is vector R cosine Theta comma R sine Theta comma 0

And then the magnitude of the cross product:

Magnitude of cross product of derivatives of S with respect to Theta and Z is R

Finally, we integrated the magnitude over the parameter domain. The result did indeed agree with the geometric formula:

Surface area equals integral from 0 to H of integral from 0 to 2 Pi of R, or 2 Pi R H

Use Mathematica to find the surface area of the parabolic “hill” parameterized as r(t,Θ) = ⟨ t cosΘ, t sinΘ, 1-t2 ⟩. Plot it first if you want to see what the surface looks like.

We didn’t have time to do much with this, other than look at the plot. You can download the Mathematica notebook that produced this plot (and more):

Plot of inverted 3D paraboloid, and a Mathematica ParametricPlot3D command that produced it

We’ll calculate the surface area in class Monday.

Next

Look at using Mathematica with surface area calculations, specifically the hill.

A bit about (scalar) surface integrals if there’s time.

No required reading, but scalar surface integrals are “Surface Integral of a Scalar-Valued Function” in section 5.6 of the textbook if you want to look at it.

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