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An introduction to the theory of integration on the real line.   FREE PDF FILE DOWNLOADS

Suitable for calculus students?   Well, suitable for an honors course in calculus.   The idea of the text is to introduce the standard material of an undergraduate course in integration theory in an unusual, but accessible, manner.  The reader will achieve an understanding of the Newton integral, the Riemann integral, and [at a rudimentary level] the Lebesgue integral.

Download the NO ANSWERS version.   This is suitable for a Moore method course---you are given the material but you need to find the proofs yourself.

Download the version complete with nearly all ANSWERS.

The Calculus Integral

Authored by Brian S Thomson

Also available for purchase on Amazon.com.

This is not classroom tested and so may need occasional corrections.  Users should  feed back any suggestions and corrections to me at thomson@sfu.ca.

We get many visitors to this site who are seeking calculus materials but find here, instead, rather more severe real analysis textbooks.   This text, The Calculus Integral,  is at the level of an honours calculus text and discusses the simple integration theory on the real line that arises from using the original Newton integral.  This integral is, arguably, a better teaching integral than the Riemann integral and better prepares the student for the modern theories of integration.

If you are a student of the calculus and just seeking other textbooks that cover the material of the standard calculus course you might find this too unusual.  But you can read it, nonetheless, with this viewpoint:  your course uses the Riemann integral and the improper Riemann integral --this course uses just the simpler calculus integral.   Any serious student of mathematics should be able to manage thinking about two closely related integration theories at the same time.   You will have to do this eventually anyway.

Note for fans of the DRIP program:  This text adopts the point of view of the DRIP program that the Riemann integral should be dumped from undergraduate instruction excepting for historical context.  Rather than presenting immediately the correct theory of integration on the real line, we use the calculus integral as a warm-up.  The student learns all the essentials of the integral including the full theory of the Riemann integral itself and an introduction to the Lebesgue integral.  The actual definition of the Riemann integral is not used to develop any of the theory.

Did you read an earlier version of this text?   If so you will notice that this version includes only the first four chapters.   A full account of integration theory on the real line that presents all the details of the Lebesgue and Henstock-Kurzweil integrals is available now on the Theory of the Integral page on this site.  It can be considered a sequel, extending these four chapters.


"Calculus Integral" or "Integral Calculus?"    The integral calculus is the name used for the subject taught in freshman calculus courses.   Most often it starts with Riemann's definition of an integral (from the mid-nineteeth century) and then introduces an "improper" version of that integral and connects (often rather clumsily) the integral with the differentiation theory.  The "calculus integral" is the integral that was used by earlier mathematicians.  It is a better starting point for learning integration theory and leads in a logical and natural manner to the more modern theories of integration on the real line that are used by all professional mathematicians today.