Chapter 5: INTEGRAL CALCULUS

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Chapter 5 INTEGRAL CALCULUS

• In Chapter 2 we used the tangent and velocity
  problems to introduce the derivative, which is
  the central idea in differential calculus. In
  much the same way, this chapter starts with the
  area and distance problems and uses them to
  formulate the idea of a definite integral, which
  is the basic concept of integral calculus.
• There is a connection between integral calculus
  and differential calculus. The Fundamental
  Theorem of Calculus relates the integral to the
  derivative, and we will see in this chapter that
  it greatly simplifies the solution of many
  problems.

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The Area Problem Find the area of the following
region
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The Definite Integral
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Evaluating Integrals
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Properties of the Definite Integral
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The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is
appropriately named because it establishes a
connection between the two branches of calculus
differential calculus and integral calculus.
Differential calculus arose from the tangent
problem, whereas integral calculus arose from a
seemingly unrelated problem, the area problem.
Newtons teacher at Cambridge, Isaac Barrow
(16301677), discovered that these two problems
are actually closely related. In fact, he
realized that differentiation and integration are
inverse processes. The Fundamental Theorem of
Calculus gives the precise inverse relationship
between the derivative and the integral. It was
Newton and Leibniz who exploited this
relationship and used it to develop calculus into
a systematic mathematical method. In particular,
they saw that the Fundamental Theorem enabled
them to compute areas and integrals very easily
without having to compute them as limits of sums
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Differentiation and Integration as Inverse
Processes
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Importance of The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is
unquestionably the most important theorem in
calculus and, indeed, it ranks as one of the
great accomplishments of the human mind. Before
it was discovered, from the time of Eudoxus and
Archimedes to the time of Galileo and Fermat,
problems of finding areas, volumes, and lengths
of curves were so difficult that only a genius
could meet the challenge. But now, armed with the
systematic method that Newton and Leibniz
fashioned out of the Fundamental Theorem, we will
see in the chapters to come that these
challenging problems are accessible to all of us.
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Indefinite Integrals or Antiderivatives
You should distinguish carefully between definite
and indefinite integrals. A definite integral
is a number, whereas an indefinite
integral is a function (or family
of functions).
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Table of Indefinite Integrals
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Applications of The Net Change Theorem
The Net Change Theorem The integral of a rate of
change is the net change
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Substitution Rule
The Substitution Rule
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Symmetry in Definite Integral
Integrals of Symmetric Functions
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The Logarithm Defined as an Integral
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Laws of Logarithms
Definition
Definition The general logarithmic function
with base is the function defined by
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The Exponential Function
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Definition
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