CHAPTER
4
4.1
4.2
4.3
4.4
CHAPTER
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
CHAPTER
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
CHAPTER
7
7.1
7.2
7.3
7.4
7.5
CHAPTER
8
8.1
8.2
8.3
8.4
8.5
Contents
The
Chain
Rule
Derivatives by the Chain Rule
Implicit Differentiation and Related Rates
Inverse Functions and Their Derivatives
Inverses of Trigonometric Functions
Integrals
The Idea of the Integral
177
Antiderivatives
182
Summation vs. Integration
187
Indefinite Integrals and Substitutions
195
The Definite Integral
201
Properties of the Integral and the Average Value
206
The Fundamental Theorem and Its Consequences
213
Numerical Integration
220
Exponentials
and
Logarithms
An Overview
228
The Exponential
ex
236
Growth and Decay in Science and Economics
242
Logarithms
252
Separable Equations Including the Logistic Equation
259
Powers Instead of Exponentials
267
Hyperbolic Functions
277
Techniques
of Integration
Integration by Parts
Trigonometric Integrals
Trigonometric Substitutions
Partial Fractions
Improper Integrals
Applications
of
the
Integral
Areas and Volumes by Slices
Length of a Plane Curve
Area of a Surface of Revolution
Probability and Calculus
Masses and Moments
8.6
Force, Work, and Energy
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C H A P T E R
Integrals
5.1
The Idea
of
the Integral
This chapter is about the idea of integration, and also about the technique of integ
ration. We explain how it is done in principle, and then how it is done in practice.
Integration is a problem of adding up infinitely many things, each of which is infini
tesimally small. Doing the addition is not recommended. The whole point of calculus
is to offer a better way.
The problem of integration is to find a limit of sums. The key is to work backward
from a limit of differences (which is the derivative). We can integrate v(x) ifit turns
up as the derivative of
another function f(x). The integral of
v
=
cos x is
f
=
sin x. The
integral of
v
=
x is
f
=
$x2. Basically, f(x) is an "antiderivative". The list of j ' s will
grow much longer (Section
5.4
is crucial). A selection is inside the cover of this book.
If we don't find a suitablef(x), numerical integration can still give an excellent answer.
I could go directly to the formulas for integrals, which allow you to compute areas
under the most amazing curves. (Area is the clearest example of adding up infinitely
many infinitely thin rectangles, so it always comes first. It is certainly not the only
problem that integral calculus can solve.) But I am really unwilling just to write down
formulas, and skip over all the ideas. Newton and Leibniz had an absolutely brilliant
intuition, and there is no reason why we can't share it.
They started with something simple. We will do the same.
SUMS A N D DIFFERENCES
Integrals and derivatives can be mostly explained by working (very briefly) with sums
and differences. Instead of functions, we have
n
ordinary numbers. The key idea is
nothing more than a basic fact of algebra. In the limit as n
+
co,
it becomes the basic
fact of calculus. The step of "going to the limit" is the essential difference between
algebra and calculus! It has to be taken, in order to add up infinitely many
infinitesimalsbut
we start out this side of it.
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 Fall '08
 PETRINA
 Fundamental Theorem Of Calculus, ax

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