ANTIDERIVATIVES
(INTEGRAL)
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8/19/11
GENERALIZED POWER FORMULA
(Integration by Simple Substitution)
OBJECTIV that can be integrated using
identify an integrand
simple
ES: substitution;using the generalized power
perform i
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TECHNIQUES OF INTEGRATION
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TECHNIQUES OF INTEGRATION
1. Integration by
parts
2. Integration by trigonometric
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substitution
3. Integration by miscellaneous
substitution
4. Integr
T OPI C
TECHNIQUES OF INTEGRATION
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TECHNIQUES OF INTEGRATION
1. Integration by
parts
2. Integration by trigonometric
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substitution
3. Integration by miscellaneous
substitution
4. Integr
TOPIC
TECHNIQUES OF INTEGRATION
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TECHNIQUES OF INTEGRATION
1. Integration by
parts
2. Integration by trigonometric
substitution
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3. Integrationsubtitle miscellaneous
substitution
4. Integrati
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APPLICATIONS
AREA subtitle style
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The area under a curve
Let us first consider
the irregular shape
shown opposite.
How can we find the
area A of this shape?
The area under a curve
We can find an
approximation by
p
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APPLICATIONS
Click to VOLUME BY INTEGRATION
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OBJECTI V
E
define what a solid of revolution is
decide which method will best determine the
volume of the solid
apply the different integration formulas.
DEFINITION
A solid of
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APPLICATIONS
CENTRIODS OF PLANE AREA
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DISCUSSI
ON
The mass of a physical body is a measure of the quantity of the matter in it,
whereas the volume of the body is a measure of the space it occupies.
If the mass pe
T OPI C
APPLICATIONS
CENTRIODS OF SOLIDS OF
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REVOLUTION
Center of gravity of a solid of revolution
The coordinates of the centre of gravity of a solid of
revolution are obtained by taking the moment of an
elementary disc
Force Due to
L iquid Pressure
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Force Due to Liquid Pressure
The force F on an area A at a depth h in a liquid
of density w is given by
F=whA
The force will increase if the density increases,
or if the depth increases or
TOPIC
TECHNIQUES OF INTEGRATION
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TECHNIQUES OF INTEGRATION
1. Integration by
parts
2. Integration by trigonometric
substitution
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3. Integrationsubtitle miscellaneous
substitution
4. Integrati
INTEGRATION OF
INVERSE HYPERBOLIC
FUNCTIONS
Review
Theorem 6.9.4 (p. 480)
Review
Theorem 6.9.5 (p. 481)
Review
Theorem 6.9.6 (p. 481)
Comparison of Integrals
(Inverse Trig & Inverse Hyperbolic)
1.
du
a u
2
= sin
2
1
u
+C
a
du
1
u
2. 2
= tan 1 + C
a
a + u2
ANTIDERIVATIVES
(INTEGRAL)
THE DEFINITE INTEGRAL
BJECTIVES:
define and interpret definite integral,
identify and distinguish the different properties of
the definite integrals; and
evaluate definite integrals
E DEFINITE INTEGRAL
If F(x) is the integral of
INTEGRALS YIELDING
THE NATURAL
LOGARITHMIC
FUNCTION
INTEGRALS YIELDING THE NATURAL
LOGARITHMIC FUNCTION
BJECTIVES:
Define the natural logarithmic function;
Illustrate solutions on problem yielding to
natural
logarithmic function;
Derive particular formu
INTEGRATION OF
EXPONENTIAL
FUNCTION
INTEGRATION OF EXPONENTIAL FUNCTION
BJECTIVES:
Define exponential functions;
Illustrate an exponential function;
Differentiate exponential function from other
transcendental function function ;
provide correct solut
ANTIDERIVATIVES
(INTEGRAL)
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8/19/11
THE BASIC TRIGONOMETRIC
INTEGRATION FORMULAS
OBJECTIVES:
recall and apply the different trigonometric identities in simplifying a
function; and
integrate trigonometric functions usin
ANTIDERIVATIVES
(INTEGRAL)
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8/19/11
TRANSFORMATION OF
TRIGONOMETRIC FUNCTIONS
OBJECTIVES:
integrate functions of the nth power of the different trigonometric
functions; and
correlate the basic trigonometric integration
WALLIS FORMULA
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8/19/11
TRIGONOMETRIC TRANSFORMATION
WALLIS FORMULA
OBJECTIVES:
recall and apply the different trigonometric identities in transforming
powers of sine and cosine; and
use Wallis Formula to shorten the s
INTEGRATION OF
INVERSE TRIGONOMETRIC
FUNCTIONS
Review
Recall derivatives of inverse trig functions
d
1 du
1
sin u =
, u <1
dx
1 u 2 dx
d
1 du
1
tan u =
dx
1 + u 2 dx
d
1
du
1
sec u =
, u <1
dx
u u 2 1 dx
2
Integrals Using Same Relationships
du
u
a 2 u 2
TOPIC
Integration of Hyperbolic
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Functions
OBJECTIVES
identify the different hyperbolic
functions;
find the integral of given hyperbolic
functions;
determine the difference between the
integrals of hyperbolic functions;
I nte
gral
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n
n
Cby a spring
alculus
Work done
Work doneby pum
ping a liquid
Work done by a spring
H ookesHookesLaw statesthat within the
Law
lim of e ticity thedisplace e produce in a
its las
m nt
d
body is proportional