Unit 8: Geometric Applications of the Integral
We consider in this unit several things that can be done with the
definite integral. In unit 7, we defined
where
as a way of calculating the area under a curve. We now extend this
idea to using integrals for other geometric purposes. In each case, we
show the quantity to be calculated can be approximated by a sum of
``slices'' of one form or another. We then take a limit process to
reach infinitely many thin slices, which will be an integral.
Objectives
After completing this unit you should be able to set up integrals to
calculate areas, areas between curves, volumes of solids of
revolutions (by both disk and shell method), arc lengths, and areas of
surfaces of revolution.
Suggested Procedure
- Simmons 7.1-7.6
- Work many of the problems at the ends of each section. Be sure
that you are capable of setting up the integral for any of the
geometric problems presented. The ability to solve such integrals is
also important, with the caveat that you haven't been given all of the
techniques of integration yet. Still, the problems are mostly designed
so as to produce an integral you can solve.
- Take the Practice
Test, Xdvi or PDF.
Independent Study page |
Calculus page
World Web Math top page
watko@mit.edu
Last modified August 1, 1998