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Integration (11.1 – 11.4)
11.1 The definite integral.
11.2 The relation between the integral and the primitive function.
11.3 Approximate methods: trapezoidal rule and Simpson’s rule.
11.4 Applications of integration: areas and volumes of solids of revolution.” (syllabus)
For a more detailed description of the requirements for this topic, see the mathematics syllabus on the Board of Studies website click here
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PRIMITIVE FUNCTIONS
We can find the derivative of a function by
applying the rule 
This gives the gradient of the function.
Can we go the other way? If we know the gradient of a function, can we find the function known as the primitive function?
Examine the gradient functions of the following.
f(x) = 5x2 + 2x + 3 f ’(x) = 10x + 2
f(x) = 5x2 + 2x + 2 f ’(x) = 10x + 2
f(x) = 5x2 + 2x + 98 f ’(x) = 10x + 2
f(x) = 5x2 + 2x + b f ’(x) = 10x + 2
f(x) = 5x2 + 2x + a f ’(x) = 10x + 2
Since all constants have a derivative of zero, all the above functions have the same gradient.
To find the primitive function of axb
we try the reverse of differentiation i.e. we raise x to a power of 1 greater
and then divide by the new power of x. hence the primitive function of axb
is 
. This means that 
= axb.
But + 4 = axb and + 80 = axb and + k = axb.
Consequently, when we find the primitive
function of axb as we also have to add a
constant. Hence the primitive function of axb is + c where c is a constant.
In some problems we are given more information such as a point on the graph so that we can evaluate this constant.
Example1: Find the primitive function when f ’(x) = 3x2 + 8x
Answer 1: f(x) = 
= 
= x3 + 4x2
+ c (don’t forget the c)
Example
2: Find the primitive function when f ’(x) = 3x -4
Answer 2: f(x) = 
= 
= -x -3 + c
or 
Exercise: Find the primitive functions of the following:
(i) x3 (ii) x + 4 (iii) x2 + 3x + 8 (iv) x5 + 9x2 (v) 2x2 + 5x + 3 (vi) x -2 (vii) 3x -2
(viii) 
(ix) 5x -3 (x) 
Answers: (i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
+ c (vii) 
(viii) 
(ix) 
(x) 
AREA OF A TRAPEZIUM
Example: Draw the line y = 2x + 2 on the number plane.
Show the points (1, 4) and (3, 8)
(i) Find the area between the line y = 2x +
2 and the x-axis between x = 1 and x = 3 by drawing a trapezium and using the
formula 
where h1
and h2 are the heights 1 & 2 and w is the width.
(ii) Find the area between the line y = 2x
+ 2 and the y-axis between x = 1 and x = 3 by drawing a trapezium and using the
formula where h1
and h2 are the heights 1 & 2 and w is the width.
(iii) Verify your answer (i) by integration.
(iv) Verify your answer (ii) by integration.
Answer:
(i) = 
= 12 units2
(ii) = 
= 8 units2
(iii) A = 
= 
= 
= (9 + 6) – (1 + 2) = 15 – 3 = 12 units2
(iv) y = 2x + 2 2x = y – 2 x = ˝ y – 1
A = 
= 
= 
= (16 – 8) – (0) = 8
units2
Exercise 1: Find the area bounded by the line y = x + 6 and (i) the x-axis and (ii) the y-axis between the points where x = 2 and x = 4 by finding the area of the trapezium. Verify your answers by integration.
Exercise2: Find the area bounded by the line y = 3x + 1 and (i) the x-axis and (ii) the y-axis between the points where x = 1 and x = 4 by finding the area of the trapezium. Verify your answers by integration.
Answers: 1. (i) 18 units2 (ii) 6 units2 2: (i) 25.5 units2, (ii) 23.5 units2
TRAPEZOIDAL RULE
Trapezoidal
Rule: A = 
{ f(a) + f(b) +
2f(a+h) + 2f(a+2h) + …….+ 2f(n-1)h }
Find the area of the first quadrant of the circle x2 + y2 = 4 by
(i)
The formula A = 
(ii) The Trapezoidal Rule using 8 intervals
________________________
(i) A = radius = 2 units
A
= 
= p units2
3.1416 units2
_________________________
8 intervals require 9 values.
Both the Trapezoidal and Simpson’s rules
include h where h = 
where b = final x-value, a = initial x-value and n = number of intervals.
It is the width of each interval.
In the above example, h = 
= 0.25
We can now draw up a table of the x-values and corresponding y-values for the beginning, end and dividing lines in between by substituting the x-values of a, a+h,
a + 2h, etc. in the equation x2 + y2 = 4.
|
|
a |
a + h |
a + 2h |
a + 3h |
a + 4h |
a + 5h |
a + 6h |
a + 7h |
b |
|
x |
0 |
0.25 |
0.5 |
0.75 |
1.0 |
1.25 |
1.5 |
1.75 |
2.0 |
|
y |
2 |
1.984 |
1.936 |
1.854 |
1.732 |
1.561 |
1.323 |
0.968 |
0 |
|
|
T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
T8 |
T9 |
(ii) Trapezoidal Rule: A = { f(a) + f(b) +
2f(a+h) + 2f(a+2h) + …….+ 2f(n-1)h }
A = 
{ 2 + 0 + 2 ( 1.984 + 1.936
+ 1.854 + 1.732 + 1.561 + 1.323 + 0.968)}
=
{ 2 + 2x11.358} = { 2 + 22.716} = 
{ 24.716}
= 3.090 units2
___________________________
Exercise: Use the Trapezoidal rule taking 6 intervals to find the area under the following curves between x = 0 and x = 3:
(i) y = x2 + 2 (ii) y = 2x2 – 3 (iii) y = 2x2 + 4x + 5
Answers: (i) 15.125 units2 (ii) 9.25 units2 (iii) 51.25 units2
SIMPSON’S RULE
Simpson’s Rule: A = 
{T1 + Tn + 2
Todd + 4Teven }
Find the area of the first quadrant of the circle x2 + y2 = 4 by using Simpson’s Rule with 8 intervals.
A = {T1 + Tn + 2Todd + 4Teven }
|
|
a |
a + h |
a + 2h |
a + 3h |
a + 4h |
a + 5h |
a + 6h |
a + 7h |
b |
|
x |
0 |
0.25 |
0.5 |
0.75 |
1.0 |
1.25 |
1.5 |
1.75 |
2.0 |
|
y |
2 |
1.984 |
1.936 |
1.854 |
1.732 |
1.561 |
1.323 |
0.968 |
0 |
|
|
T1 |
T2 |
T3 |
T4 |
T5 |
T6 |
T7 |
T8 |
T9 |
= 
{ 2 + 0 + 2(1.936 + 1.732 + 1.323) + 4(1.984 + 1.854 + 1.561
+ 0.968)}
= { 2 + 2x4.991 + 4x6.367 }
{ 2 + 9.982 + 25.468 }
{37.45 }
= 
{37.45}
= 3.121 units2
__________________________________________________
Exercise: Use Simpson’s rule, taking 6 intervals to find the area under the following curves between x = 0 and x = 3:
(i) y = x2 + 2 (ii) y = 2x2 – 3 (iii) y = 2x2 + 4x + 5
Answers: (i) 15.0 units2 (ii) 9.0 units2 (iii) 51.0 units2