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THE INTEGRAL CALCULUS
Alan Smithee
For information about this publication consult the web site
classicalrealanalysis.com
Direct all correspondence to Alan Smithee
calculusprogram at gmail.com
1991 Mathematics Subject Classiﬁcation. Primary 26A06, 26A24, 26A39, 26A42;
Secondary 26A45, 26A46,
Key words and phrases. Derivative, integral, Newton integral, Henstock-Kurzweil
integral.
The author has asked for these notes to be published on the web site
www.classicalrealanalysis.

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THE INTEGRAL CALCULUS
Alan Smithee
For information about this publication consult the web siteclassicalrealanalysis.comDirect all correspondence to Alan Smitheecalculusprogram
at
gmail.com
1991
Mathematics Subject Classiﬁcation.
Primary 26A06, 26A24, 26A39, 26A42;Secondary 26A45, 26A46,
Key words and phrases.
Derivative, integral, Newton integral, Henstock-Kurzweilintegral.The author has asked for these notes to be published on the web sitewww.classicalrealanalysis.comand made available without charge. They can be used and distributed byeducators or students of the calculus, but not altered without permission.Copyright assigned to the authors of www.classicalanalysis.com.
VERSION 0.22 [BETA]October 24, 2006.
Direct correspondence to Alan Smithee .
Abstract.
The D.R.I.P. program (
DUMP the RIEMANN INTEGRAL
)proposes that the usual calculus treatment of the Riemann integral and theintroduction of so-called improper integrals be dropped in favor of a simplerintegral which includes those theories. These notes suggest a way to do this.
Preface
These informal calculus notes are oﬀered to those instructors who may wish todesign a calculus sequence that drops the Riemann integral in favor of the morenatural integral on the real line that has been called, at diverse times, by manydiﬀerent names
1
.The program to drop the classical Riemann integral from the undergraduatecurriculum has been called by some
D.R.I.P.
[Dump the Riemann Integral Project]and has a small and mostly ignored band of followers. Encouraged by the authorsof the web sitewww.classicalrealanalysis.comI have made these notes freely available to anyone wishing to use them for these pur-poses. I am grateful to these three authors, but absolve them of any responsibilityfor the many ﬂaws of this presentation.I am not the srcinator of these ideas, merely a scribe who has undertakento deliver and interpret some notes that fell into my possession under unusualcircumstances. These will be related in a later publication
The Fundamental Program of the Calculus
,by Alan Smithee (2007)
[to appear].
The notes are not classroom tested. Indeed I am far too timid to try themout. My chairman would have my hide if I were to teach our calculus studentsanything other than the Riemann integral and the improper integral from the stan-dard prescribed texts. Moreover, should I try covertly to expose the students in mysections to these ideas and these simpler methods, they might fail in the generalexaminations. For example, my students would arrive at the formula
ba
x
p
dx
=
b
p
+1
−
a
p
+1
p
+ 1 (
p >
0)by the simple justiﬁcation of the diﬀerentiation formula
ddxx
p
+1
/
(
p
+ 1) =
x
p
.
But this would be graded as incomplete because traditional calculus students (usingthe dreadful Riemann integral) are obliged also to verify that the function
f
(
x
) =
x
p
is in fact Riemann integrable. The very fact that it is a derivative of another
1
The Denjoy integral, the Perron integral, the Denjoy-Perron integral, the restricted integralof Denjoy, the Denjoy total, the Henstock integral, the Kurzweil integral, the Henstock-Kurzweilintegral, the Kurzweil-Henstock integral, the generalized Riemann integral, the Riemann-completeintegral, the gage integral, the gauge integral, etc. We call it simply
the integral
.
iii
iv PREFACE
function is adequate justiﬁcation in the correct theory of integration, but not at allfor the Riemann integral.More distressingly would be their solution to this problem: Verify that
10
x
p
dx
= 11 +
p
(
−
1
< p <
0)
.
My students would use the same diﬀerentiation formula (valid for 0
< x
≤
1) andmerely point out that the function
F
(
x
) =
x
p
+1
p
+ 1is continuous on [0
,
1] provided that
−
1
< p <
0. They would then claim that
10
x
p
dx
=
F
(1)
−
F
(0) = 11 +
p
(
−
1
< p <
0)
.
Again wrong? Few of my colleagues would understand that this is perfectly correct;they would demand instead some nonsense about unbounded functions, improperintegrals, and limits on the right-hand side at the endpoint 0.But if these worries do not alarm you then, certainly, try out the notes. But letme know of the successes and failures as well as improvements that can be madein the notes.Alan SmitheeDirect correspondence to Alan Smithee

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