Calculus : Integration

1
Calculus Integration

• Part 1

2
What we will do

• Integration as the Process of Summation
• Integration as the Reverse of Differentiation
• Techniques of Integration
• Integration by substitution
• Integration by parts
• Integration by partial fractions
• Integration of special functions

3
What we will do

• Some Consequences of the Fundamental Theorem of
  Calculus
• Applications of Integration Areas Bounded by
  Curves and Volumes of Revolution
• Reduction Formula

4
Integration Concept and Theory

• We know how to find the area of simple geometric
  shapes such as the triangle below

5
Integration Concept and Theory

• But how do we find the are of geometric object
  which do not have straight edges ?

6
Integration Concept and Theory

• So, how do we go about finding the area under the
  curve f(x), between xa and xb ?
• Well,
• we can divide the area under the curve into
  separate rectangles
• find the area of each rectangle
• and then sum these areas in order to find an
  approximate answer to area under curve

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Integration Concept and Theory

• Find area of each rectangle
• then sum all areas between xa and xb

8
Integration Concept and Theory

• Example
• A10.5
• A20.625
• A31
• A41.625
• Total A 3.75
• exact A 4.67

9
Integration Concept and Theory

• Example
• A10.625
• A21
• A31.625
• A42.5
• Total A 5.75
• exact A 4.67

10
Integration Concept and Theory

• Problem answers are too small of too big. How
  to improve result ?
• Answer make strips thinner

11
The Fundamental Theorem of Calculus

• Consider 1 very thin strip height yi width dx

12
The Fundamental Theorem of Calculus

• Then
• between xa and xb (i1 n)

13
The Fundamental Theorem of Calculus

• In the limiting process (i ? infinity)

14
The Fundamental Theorem of Calculus

• Graphically

15
The Fundamental Theorem of Calculus

• So finding area under curve y gt integration
• and this is expressed symbolically as
• this is one part of the fundamental theorem of
  calculus

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The Fundamental Theorem of Calculus

• We now know integration concept of area
• but how to actually perform integration ?
• It can be shown that integration can be performed
  as the reverse of differentiation

17
The Fundamental Theorem of Calculus

• To see this consider curve with variable x coord
• Then there is a function A(x) which area under
  curve so far

18
The Fundamental Theorem of Calculus

• Consider now the sequence of areas under yx2
• Between x0 and x1 area 0.333333
• Between x0 and x2 area 2.666666
• Between x0 and x3 area 9
• Etc
• This sequence of numbers happens to be the answer
  to evaluating an area function A(x)

19
The Fundamental Theorem of Calculus

• From previous fundamental theorem of calculus we
  have
• i.e. there is a function which represents area
  under curve y

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The Fundamental Theorem of Calculus

• Consider extending this theorem to find the area
  between 0 and xdx
• and let dx ? x

21
The Fundamental Theorem of Calculus
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The Fundamental Theorem of Calculus

• We then consider the elemental area
• hence

23
The Fundamental Theorem of Calculus

• then
• this is the other part of the fundamental theorem
  of calculus

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The Fundamental Theorem of Calculus

• This means that as dx ? x we recover the function
  A(x) from the 1st part of the theorem
• hence A (the answer we are looking for) is
  obtained by finding a function which
  differentiates to give y