Antidifferentiation: The Indefinite Intergral Chapter Five.
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Transcript of Antidifferentiation: The Indefinite Intergral Chapter Five.
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Antidifferentiation: The Indefinite Intergral
Chapter Five
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§5.1 Antidifferetiation
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§5.1 General Antiderivative of a Function
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§5.1 General Antiderivative of a Function
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§5.1 Rules for Integrating Common Function
The Constant Rule
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§5.1 Rules for Integrating Common Function
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Example:
Solution:
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§5.1 Applied Initial Value Problems
An initial Value problems is a problem that involves solving a differential equation subject to a specified initial condition. For instance, we were required to find y=f(x) so that
A Differential equation is an equation that involves differentials or derivatives.
We solved this initial problem by finding the antiderivative
And using the initial condition to evaluate C.
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The population p(t) of a bacterial colony t hours after observation begins is found to be change at the rate
If the population was 2000,000 bacteria when observations began, what will be population 12 hours later?
Example:
Solution:
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§5.2 Integration by Substitution
How to do the following integral?
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§5.2 Integration by Substitution
Think of u=u(x) as a change of variable whose differential is
Then
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Example:
Solution:
Find
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Example:
Solution:
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Example:
Solution:
To be continued
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Example:
Solution:
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Example:
Solution:
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§5.3 The Definite Integral and the Fundamental Theorem of Calculus
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All rectangles have same width.
• n subintervals:
• Subinterval width
•Formula for xi:
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• Choice of n evaluation points
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Right-endpoint approximation
left-endpoint approximation
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Midpoint Approximation
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Example:
=0.285
To be continued
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=0.3325
=0.385
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Example:
left-endpoint approximation
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Midpoint Approximation
Right-endpoint approximation
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00=1.098608585 =1.098611363
Area Under a Curve Let f(x) be continuous and satisfy f(x)≥0on the interval a≤x≤b. Then the region under the curve y=f(x)over the interval a≤x≤b has area
1 21
lim lim[ ( ) ( ) ... ( )] lim ( )n
n n jn n n
j
A S f x f x f x x f x x
Where xj is the point chosen from the jth subinterval if the Interval a≤x≤b is divided into n equal parts, each of length
b ax
n
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§5.3 The Definite Integral
Riemann sum Let f(x) be a function that is continuous onthe interval a≤x≤b. Subdivide the interval a≤x≤b into n equal
parts, each of width ,and choose a number xk from the
kth subinterval for k=1, 2, …, . Form the sum
b ax
n
Called a Riemann sum.
Note: f(x)≥0 is not required
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§5.3 The Definite Integral
The Definite Integral the definite integral of f on the interval
a≤x≤b, denoted by , is the limit of the Riemann sum asn→+∞; that is
b
af(x)dx
The function f(x) is called the integrand, and the numbers a and b are called the lower and upper limits of integration, respectively. The process of finding a definite integral is called definite integration.
Note: if f(x) is continuous on a≤x≤b, the limit used to define integral exist and is same regardless of how the subinterval representatives xk are chosen.
b
af(x)dx
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§5.3 Area as Definite Integral
If f(x) is continuous and f(x)≥0 for all x in [a,b],then
( ) 0b
af x dx
and equals the area of the region bounded by the graph f and the x-axis between x=a and x=b
If f(x) is continuous and f(x)≤0 for all x in [a,b],then
( ) 0b
af x dx
And equals the area of the region bounded by the graph f and the x-axis between x=a and x=b
( )b
af x dx
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§5.3 Area as Definite Integral
equals the difference between the area under the graph
of f above the x-axis and the area above the graph of f below the x-axis between x=a and x=b
This is the net area of the region bounded by the graph of f and the x-axis between x=a and x=b
( )b
af x dx
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§5.3 The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus If the function f(x) is continuous on the interval a≤x≤b, then
( ) ( ) ( )b
af x dx F b F a
Where F(x) is any antiderivative of f(x) on a≤x≤b
Another notation:
( ) ( ) | ( ) ( )b b
aaf x dx F x F b F a
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§5.3 The Fundamental Theorem of Calculus (Area justification )
In the case of f(X)≥0, represents the area the curve y=f(x) over the interval [a,b]. For fixed x between a and b let A(x) denote the area under y=f(x) over the interval [a,x].
( )b f x dxa
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By the definition of the derivative,
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Differentiation
Indefinite Integration
Definite integration
Example
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§5.3 Integration Rule
Subdivision Rule
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§5.3 Subdivision Rule
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Example
Solution:
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Example
Solution:
To be continued
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§5.3 Substituting in a definite integral
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23
3
222 3
300
1 1 2 21
3 3 31
2 41
3 31
xdx du u x
ux
xdx x
x
2.
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§5.3 Substituting in a definite integral
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Example
Solution:
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Example
Solution:
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