A Course in Mathematics for Students in Engineering and Applied Science. Volume 2

432 Pages·44.281 MB·English

by  Woods F., Bailey F.

APR 191909
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A COU R E IN
MATHEMATICS
FOR STUDENTS OF HNGINEERING AXD
APPLIED SCIBNCE
BY
FUEDERICK S. WOODS
A~D
FREDERICK H. BAILEY
I'twn:~sons OF :IIATIIEliATtcs t:< Tnt! )IASSAcnusBTTs
l:<STITUT& OF 'l'&CIINOLOOY
VoLmiE II
INTEGRAL CALCULUS
FUNCTIONS OF SEVERAL VARIABT.ES, SPACE GEOMETRY
DIFFERENTIAL EQUATIONS
GINN & COMPANY
BOSTON · NEW YORK · CHICAGO • LONDON
COPVlll(lBT, 1009, BY
Fnrm&lliCK S. WOODS AND FRF:DERJCK H. BAJLEY
ALL RIOBTS RESERVED
Gl:b• !:ltbon,.um llu•lt
GINN & COMPANY • PR.O·
PR.IETORS • DOSTON • U.S.A.
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PREFACE ¥-
Tlll.s ,·olume completes the plan of a course in mathematics out
lined in the preface to the first ,·olume.
The subject of integration of functions of a single variable is
treated in the first eight chapters. Emphasis is here laid upon the
fundamental processes, and the conception of the definite integral
and its numerous applications are early introduced. Only after the
student is well grounded in these matters are the more special
methods of evaluating integrals discussed. In this way the student's
interest is early aroused in the use of the subject, and he is drilled
in tbo$8 processes which occur in subsequent practice. A new
feature of this part of the book is a chapter on simple differential
equations in close connection with integration and long before the
formal study of differential ec1uati~~·;
With the ninth chapter the study of functions of two or more
variables is begun. This is introduced and accompanied by the use
of the elements of solid analytic geometry, and the treatment of
partial differentiation and of multiple integrals is careful and rea
sonably complete. A special feature here is the chapter on line
integrals. This subject, though generally omitted from elementary
texts, is needed by most engineering students in their later work.
The latter part of the work consists of chapters on series, the
complex number, and differential equations. In the treatment of
differential equations many things properly belonging to an ex
tended treatise on the subject are omitted in order to giYe the
student a concise working knowledge of the types of equations
which occur most. often in practice.
iii
M298'790
lV PREFACE
In conclusion the authors wish to renew their thanks to the
members of the mathematical uepartmeot of the Massachusetts
Institute of Technology, and especially to Professor H. W. Tyler,
for continued helpful suggestion and criticism, and to extend
thanks to their former colleague, Professor W. H. Roever of
'Vashington University, for the construction of the more difficult
drawings particularly in space geometry.
?lf.USACIIUSt:TTS IXSTITUTE OF 'l't:Clll'OLOOY
February, 1909
CONTENTS
CIIAl'TER I- I~FINlTESDIALS A~D DIFFERE:\TIALS
ARTICLE
1-2. Order of infinitesimals . . . .
3. Fundamental theorems on infinitesimal~ 4
4. Differentials . . . . . 6
6-6. Graphical representation . . 8
7. Formulas for differentials 9
8. Differentials of higher orders 10
CHAPTER 11- ELE~lE~TARY FORl\fULAS OF L.~TEGRATIO~
9. Definition of integration 12
10. Constant of integration 12
11. Fundamental formulas 13
12. Integral of u• . . . . 14
13. Integrals of trigonometric functions 17
14-15. Integrals leading to the inverse trigonometric functions 19
16. Integrals of exponential functions 2.>
17. Collected formulas . . . 26
18. Integration by substitution 27
10. Integration by pa~. . 30
20. PossibiHty of integration 32
Problems . . . . . 33
CHAPTER III-DEFI:\'ITE Ii\'TEGRALS
21. Definition . . . . . 39
22. Graphical representation 41
23. Generalization 4:1
24. Properties of definite integrals 4.3
Evaluation of the definite integral by inu•gration 47
Change of limits . . 49
27. Integration by parts 51
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28. Infinite limits . 52
29. Infinite integrand 5!1
!10. The mean ''alue of a function :H
31. Taylor's and Maclaurin's series. 5G
32-33. Operations with power series GO
l'roblems . . . 61
CIIAPTF.H JY- APPLICATIO~S TO GEO:IIETRY
3·!. Element of a definite integral . 01
;15. Area of a plane curve in Cartesian coordinates Ol
36. Area of a plane curve in polar coordinates 67
37. Volume of a solid of revolution . . 60
38. Yolume of a solid with parallel bal>es. . 72
39. Tho prismoidal formula 74
40. Length of a plane curve in rectangular coordinates n
41. Ll.'ngth of a plane curve h1 polar coordinates 77
42. 'l'hc differential o( arc . 78
43. Area. of a surface of revolution . 70
Problems . . . so
CUAPTJ~R V -APPLICATIOXS TO :IIECIIANICS
44. Work 86
4.3. Attraction . . 86
40. Pressure. 88
47. Center of gravity 00
48. Center of gravity of a plane curvG . 92
49. Center of gravity of a plane arl.'a . 91
50. Center of gravity of a solid or a sw1ace of revolution of con-
stant density 06
51. Center of pressure 98
Problems 09
CHAPTER YT - INTEGRATIOX OF RATIONAL FRACTIO~S
.32. Introduction . 103
53-0 !. Separation into partial fractionil 10 L
55-57. Proof of the possibility of sepn.ration into partial fractions 100
58. Integration of rational fractions 11:\
Problems 117