substitution method integration. when one function is not the derivative of the other e.g. x is not...
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Substitution Method
Integration
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
Example 2
x - 1 is not the derivative of x +4 and it contains a variable
Substitute
Integrating and substituting back in for u
Delta Exercise 12.8
The definite integral
Example 1
As 2x is the derivative, use inverse chain rule to integrate
Substitute x = 4 Substitute x = 2
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
Example 3
Use substitution
Substituting
Delta Exercise 12.9
Areas under curves
To find the area under the curve between a and b…
…we could break the area up into rectangular sections. This would
overestimate the area.
…or we could break the area up like this which would
underestimate the area.
The more sections we divide the area up into, the more accurate our answer would be.
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.
is the sum all the areas of infinitely narrow width, dx and height, y.
As the value of dx decreases, the area of the rectangle approaches y x dx
0 dx
y
The area of this triangle is 3 units squared
30
2
The equation of the line is
dx
y
If we sum all rectangles
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
2011 Level 2
2011 Level 2
2010 Level 2
2010 Level 2
• Area cannot be negative
• Area = 6.67 units2
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
-6 -1 8
2011 Level 2
2011 Level 2
2010 Question 1c
2010 Question 1c
2012
2012
2012
2012
• First find the x-value of the intersection point
2012
2010 Question 1e
2010 Question 1e
• Find intersection points
2010 Question 1e
Looking at areas a different way
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
Areas between two curves
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
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
Example 2A typical rectangle
y - y
dx
Area = (y - y)dx
0.707 Area
Practice
More practice
Delta Exercise 16.2, 16.3, 16.4Worksheet 3 and 4
Area in polar: extra for experts