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Substitution Method
Integration
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When one function is not the derivative of the other e.g.
x is not the derivative of (4x -1) and
x is a variable
Substitute
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Example 2
x - 1 is not the derivative of x +4 and it contains a variable
Substitute
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Integrating and substituting back in for u
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Delta Exercise 12.8
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The definite integral
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Example 1
As 2x is the derivative, use inverse chain rule to integrate
Substitute x = 4 Substitute x = 2
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Example 2
Divide the top by the bottom
4x divided by 2x = 2
Solving x = 1/2 Substitute x = 1/2
into 4x + 3 to get 5
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Example 3
Use substitution
Substituting
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Delta Exercise 12.9
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Areas under curves
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To find the area under the curve between a and b…
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…we could break the area up into rectangular sections. This would
overestimate the area.
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…or we could break the area up like this which would
underestimate the area.
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The more sections we divide the area up into, the more accurate our answer would be.
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If each of our sections was infinitely narrow,
we would have the area of each section as
y
The total area would be the sum of all these areas between a and b.
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is the sum all the areas of infinitely narrow width, dx and height, y.
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As the value of dx decreases, the area of the rectangle approaches y x dx
0 dx
y
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The area of this triangle is 3 units squared
30
2
The equation of the line is
dx
y
If we sum all rectangles
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The area of this triangle is 3 units squared
30
2
The equation of the line is
dx
yIf we sum all
rectanglesThe area is 3
but the integral is -3
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2011 Level 2
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2011 Level 2
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2010 Level 2
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2010 Level 2
• Area cannot be negative
• Area = 6.67 units2
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CombinationIntegral is positive
Integral is negative
To find the area under the curve, we must integrate between -6 and -1 and between 8 and -1 separately and add the positive values together.
-6 -1 8
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-6 -1 8
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2011 Level 2
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2011 Level 2
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2010 Question 1c
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2010 Question 1c
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2012
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2012
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2012
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2012
• First find the x-value of the intersection point
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2012
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2010 Question 1e
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2010 Question 1e
• Find intersection points
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2010 Question 1e
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Looking at areas a different way
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As the value of dy decreases, the area of the rectangle approaches x x dy
0
dy
x
Definite Integral is
3
4
The equation of the line is
Rearrange
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Areas between two curves
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A typical rectangle in the upper section
x - x
dyArea =(x - x )dy
x = y
Area for this section is
1
Solving theseEquations gives
y = 1
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A typical rectangle in the lower section
x - xdyArea =(x - x )dy
x = y
Area for this section is
Total area is equal to 1
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Example 2A typical rectangle
y - y
dx
Area = (y - y)dx
0.707 Area
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Practice
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More practice
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Delta Exercise 16.2, 16.3, 16.4Worksheet 3 and 4
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Area in polar: extra for experts