Construction of mathematical models of physical phenomena requires functional domains of greater complexity than the previously employed line segments and plane regions. This section makes progressin meeting that need by enriching integral theory with the introduction of segments of curves andportions of surfaces as domains. Thus, single integrals as functions defined on curve segments take on new meaning and are then called line integrals. Stokes’s theorem exhibits a striking relation betweenthe line integral of a function on a closed curve and the double integral of the surface portion that is, enclosed. The divergence theorem relates the triple integral of a function on a three-dimensional region of space to its double integral on the bounding surface. The elegant language of vectors best describes
these concepts; therefore, it would be useful to reread the introduction to Chapter 7, where the importance of vectors is emphasized. (The integral theorems also are expressed in coordinate form.)
LINE INTEGRALS
The objective o this section is to geometrically view the domain of a vector or scalar function as a
segment of a curve. Since the curve is defined on an interval of real numbers, it is possible to refer the
function to this primitive domain, but to do so would suppress much geometric insight.
A curve, C, in three-dimensional space may be represented by parametric equations:
INTRODUCTION OF THE DEFINITE INTEGRAL
The geometric problems that motivated the development of the integral calculus (determination of
lengths, areas, and volumes) arose in the ancient civilizations of Northern Africa. Where solutions were
found, they related to concrete problems such as the measurement of a quantity of grain. Greek
philosophers took a more abstract approach. In fact, Eudoxus (around 400 B.C.) and Archimedes
(250 B.C.) formulated ideas of integration as we know it today.
Integral calculus developed independently, and without an obvious connection to differential
calculus. The calculus became a ‘‘whole’’ in the last part of the seventeenth century when Isaac Barrow,
Isaac Newton, and Gottfried Wilhelm Leibniz (with help from others) discovered that the integral of a
function could be found by asking what was differentiated to obtain that function.
The following introduction of integration is the usual one. It displays the concept geometrically and
then defines the integral in the nineteenth-century language of limits. This form of definition establishes
the basis for a wide variety of applications.
Consider the area of the region bound by y = f ( x), the x-axis, and the joining vertical segments
(ordinates) x= a and x = b
CHANGE OF VARIABLE OF INTEGRATION
If a determination of f(x) dx is not immediately obvious in terms of elementary functions, useful
results may be obtained by changing the variable from x to t according to the transformation x = gً)t).
(This change of integrand that follows is suggested by the differential relation dx = g("t( dt.) The fundamental
theorem enabling us to do this is summarized in the statement
f )x) dx = f (gt)g(t)" dt
where after obtaining the indefinite integral on the right we replace t by its value in terms of x, i.e.,
t = g(ًx). This result is analogous to the chain rule for differentiation ().
The corresponding theorem for definite integrals is
Partial fractions. Any rational function
where P)x( and Q)x( are polynomials, with the
degree of P)x( less than that of Q)x(, can be written as the sum of rational functions having the
form
,
where r = 1; 2; 3; . . . which can always be integrated in terms of
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .