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Ch 4 Integration
History Lesson!Early accomplishment of Calculus predicting the future position of a moving body from one of its known locations and a formula for its velocity function!
Today, we generalize and say recover a function from one of its known values and a formula for its rate of change!
Applications: Real life
to calculate a company's future output, from its present output and its production rate!
to predict a population's future size, from its present size and its growth rate!
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Integration
Quick Overview:
Integral Calculus the second main branch of calculus.
the process of finding a function whose derivative is known!
to give the sum or total of.
Types of Integralsdefinite
indefinite
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Antiderivatives
4.1 Antiderivatives and Indefinite Integration
Find a function F(x) whose derivative is f (x) = 2x.
F(x) = x2 because
ddx( (x2 = 2xothers?!!
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A function F(x) is an antiderivative of f (x)
on an interval I if "cap eff" "little eff"
for all x on I.
F (x) = f (x)'
Def.
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terminology and notation
The General Antiderivative of f :
F(x) + C family of all antiderivatives of f
slide up!
Cconstant of integration
A differential equation in x and y is an equation that involves x, y, and derivatives of y.
ex. y' = 4x + 1
Def.
Also, F(x) = x2 + C is the general solution of the differential equation F'(x) = 2x.
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Example: Find the general solution of the differential equation
dydx = x
2 3/.
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Notation for Antiderivativesdydx = f (x)
dy dx= f (x) differential form
antidifferentiation or indefinite integration
the operation of finding all solutions
denoted by an integral sign,
dy dx= f (x)
y = F(x) + C general solution
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(more) Notation
dx =f (x)y = F(x) + C
PRESS
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(more) Notation
dx =f (x)y = F(x) + C
integrand
variable of integrationdx identifies x as the
constant ofintegration
is read as "the antiderivative (or indefinite integral) of f with respect to x".dxf (x)
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Examples: Find the indefinite integrals. Basic Integration Rules p244
(x3 1) dx1.
2. ( +1 (
2 xx2 3x4 dx
3. sin d
4. sin x dxcos x 2
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Key Relationship!
Integration is the "inverse" of differentiation.
=dxf (x) FdxF' =(x) (x) + C
Differentiation is the "inverse" of integration.
= f (x)F(x) + Cdxd [ [ f(x)dx
d [ dx
[=
Obtain integration formulas directly from differentiation formulas
Basic Integration Rules p244
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p249-250 #1-3, 5-8, #9 - 42 (odd?!), #43 - 48
Assignment