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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
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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View Full DocumentCHAPTER
8
Applications
of
the Integral
We are experts in one application of the integralto
find the area under a curve.
The curve is the graph of y
=
v(x), extending from x
=
a at the left to x
=
b at the
right. The area between the curve and the x axis is the definite integral.
I think of that integral in the following way. The region is made up of
thin strips.
Their width is dx and their height is v(x). The area of a strip is v(x) times dx. The
area of all the strips is
1:
v(x) dx. Strictly speaking, the area of one strip is
meaninglessgenuine
rectangles have width Ax. My point is that the picture of thin
strips gives the correct approach.
We know what function to integrate (from the picture). We also know how (from
this course or a calculator). The new applications to volume and length and surface
area cut up the region in new ways. Again the small pieces tell the story. In this
chapter, what to integrate is more important than how.
8.1
Areas and Volumes
by
Slices
This section starts with areas between curves. Then it moves to
volumes,
where the
strips become slices. We are weighing a loaf of bread by adding the weights of the
slices. The discussion is dominated by examples and figuresthe
theory is minimal.
The real problem is to set up the right integral. At the end we look at a different way
of cutting up volumes, into thin shells. All formulas are collected into ajnal table.
Figure 8.1 shows
the area between two curves.
The upper curve is the graph of
y
=
v(x). The lower curve is the graph of y
=
w(x). The strip height is v(x)

w(x), from
one curve down to the other. The width is dx (speaking informally again). The total
area is the integral of "top minus bottom":
area between two curves
=
[v(x)

w
(x)] dx.
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 Fall '08
 PETRINA

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