Section 5.5 Surface Area Formulas for Implicit and Explicit Descriptions
We now turn to compute surface areas when the surface is described first implicitly and second explicitly. Turns out the explicit formula is just a quick extension of the implicit formula so up to you whether you want to learn the two formula by memory or just the implicit formula together with how to convert an explicit to implicit.
Learning Objectives:
State the formula for Surface Area of an Implicitly or Explicitly define
Determine the appropriate \(\vec{p}\) for a surface
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1.
I skipped the formal derivation of the Implicit Surface Area formula. It's not bad practice: basically we come up with a generic parameterization and then use the parameterization surface area formula to get our implicit formula. The method is found
here in the Thomas Calculus textbook used in Math 204 at UVic 1 . I do want to note that while this method is mainly computational using techniques you know, the real magic that makes it work is called the Implicit Function Theorem which is extremely important and rather deep and perhaps the biggest single thing we gloss over in our rather non-rigorous treatment of vector calculus.
media.pearsoncmg.com/cmg/pmmg_mml_shared/mathstats_html_ebooks/ThomasCalcET14e/page_1017.html
