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

http://rowdy.mscd.edu/~talmanl/MathAnim.html

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