MA101: Single-Variable Calculus I

8.5: Surface Areas of Solids

• In this subunit, we will combine what we learned earlier in this unit. Though you might expect that calculating the surface area of a solid will be as easy as finding its volume, it actually requires a number of additional steps. You will need to find the curve-length for each of the slices we identified earlier and then add them together.
  

• University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 6: Applications of the Integral, Section 6.4: Area of a Surface of Revolution" URL

  Read Section 6.4 (pages 327 through 335). In this beautiful presentation of areas of surfaces of revolution, the author again makes use of rigorously-defined infinitesimals, as opposed to limits. Recall that the approaches are equivalent; using an infinitesimal is the same as using a variable and then taking the limit as that variable tends to zero.
  

• Massachusetts Institute of Technology: David Jerison's "Parametric Equations, Arclength, Surface Area" Page

  Watch this video from 26:10 to 40:35.
  

• Clinton Community College: Elizabeth Wood's "Supplemental Notes for Calculus II: Areas of Surfaces of Revolution" URL

  Work through each of the three examples on the page. As in any assignment, solve the problem on your own first. Solutions are given beneath each example.