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The book uses classical problems to motivate a historical
development of the integration theories of Riemann, Lebesgue,
Henstock-Kurzweil and McShane, showing how new theories of
integration were developed to solve problems that earlier
integration theories could not handle. It develops the basic
properties of each integral in detail and provides comparisons of
the different integrals. The chapters covering each integral are
essentially independent and could be used separately in teaching a
portion of an introductory real analysis course. There is a
sufficient supply of exercises to make this book useful as a
textbook.
Changing the way students learn calculus at New Mexico State
University. In the Spring of 1988, Marcus Cohen, Edward D. Gaughan,
Arthur Knoebel, Douglas S. Kurtz, and David Penegelley began work
on a student project approach to calculus. For the next two years,
most of their waking hours (and some of their dreams) would be
devoted to writing projects for their students and discovering how
to make the use of projects in calculus classes not only
successful, but practical as well. A grant from the National
Science Foundation made it possible for this experiment to go
forward on a large scale. The enthusiasm of the original group of
five faculty was contagious, and soon other members of the
department were also writing and using projects in their calculus
classes. At the present time, about 80% of the calculus students at
New Mexico State University are doing projects in their Calculus
courses. Teachers can use their methods in teaching their own
calculus courses. Student Research Projects in Calculus provides
teachers with over 100 projects ready to assign to students in
single and multivariable calculus. The authors have designed these
projects with one goal in mind: to get students to think for
themselves. Each project is a multistep, take-home problem,
allowing students to work both individually and in groups. The
projects resemble mini-research problems. Most of them require
creative thought, and all of them engage the student's analytic and
intuitive faculties. the projects often build from a specific
example to the general case, and weave together ideas from many
parts of the calculus. Project statements are clearly stated and
contain a minimum of mathematical symbols. Students must draw their
own diagrams, decide for themselves what the problem is about, and
what toolsfrom the calculus they will use to solve it. This
approach elicits from students an amazing level of sincere
questioning, energetic research, dogged persistence, and
conscientious communication. Each project has accompanying notes to
the instructor, reporting students' experiences. The notes contain
helpful information on prerequisites, list the main topics the
project explores, and suggests helpful hints. The authors have also
provided several introductory chapters to help instructors use
projects successfully in their classes and begin to create their
own.
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