P1 Chapter P1 Chapter 1616
CIE Centre A-levelCIE Centre A-level Pure Maths Pure Maths
© Adam Gibson
IntegrationIntegration
Differentiation
dy
dx
Integration
ydx
SumDifference
Gradient Area
Calculus
Calculus
Integrals
Definite Integral Indefinite Integral
( )f x dx( )x b
x af x dx
Noun (Action): integration
Verb: integrateNoun: integral
Derivatives
Read it as “integral [of] f [of] x dx [between a and b]”
Integration as the reverse of differentiation
The first, and easiest, way to understand integration:What function g(x) has f(x) as its gradient at all points x in the domain?
The indefinite integral helps us find the answer.
Example:
2What function ( ) has gradient given by 2 1 ?g x y x
We remember the rule that
1 n ndyy ax nax
dx
Think….
Integration as the reverse of differentiation – contd.
We note that we can change to (I won’t give a formal proof here, but you can verify it).
323( )g x x x ! Warning …
What if 323( ) 10g x x x ?
So we construct the rule:
1
1n na
ax dx x kn
where k is an “arbitrary
constant”.This is the indefinite integral.
Implications
•Does the formula apply for all rational numbers n?
•No! At n = -1 the formula “explodes”. We will consider this in detail later.
•What does “arbitrary constant k” actually mean?
•For this point, think about moving the graph of a function up and down by adding a number. Does the gradient change?
•So there are an infinite number of functions with the same gradient formula. We can only calculate k if we are giventhe coordinates of at least one point on the graph. If you don’t have specific information, always write “+k”.
TASKSTASKS
1. Find the integral of
2. Find
3. Find
4. Try Q15 and Q16 p. 240
2 1x x
( )g x xdx given that g(1)=0
32
1x dx
x
Integration as areaarea
1x 2x 3x
Area Area of each trapeziumx
4x
How do we make it more accurate?
…
Integration as area – area – continued.continued.
Just like finding the gradient, this is a limit problem.
The exact area is given by
0lim (area of trapezium with base width x)x
x
A
We take the sum of an infinite number of trapezia, each with infinitely narrow width.
Thus, if the function is f(x), we get:
0
11
1 120
0
( ( ) ( )).( )
lim ( ).
( )n
n
k k k kk
xx
x
x
A f x f x x x
A f x x
A f x dx
Remember – dx is an infinitely small ∆x
Section 16.2 and Section 16.3 givea more detailedderivation
Integration as area – area – continued.continued.
0
2
4
6
8
10
12
0 0.5 1 1.5 2 2.5
3 2y x
Let’s find the area underthe curve between x=0 and x=2
“Limits of integration”
Integration as area – area – continued.continued.
We write this quantity as the definite integral:
2 3
0( 2)x dx
Higher limit at the top
Lower limit at the bottom
24
0
24
xx
You should perform the integration,but not include the arbitraryconstant k (can you see why?).Then you should put the answer insquare brackets with the limitsmarked on the right.
Integration as area – area – continued.continued.
24
0
24
xx
Now for the last step. Evaluatethe expression inside the bracketsat x=upper limit and x=lower limit,and subtract the latter from the former.
4 42 02 2 2 0
4 4
8
What are the unitsfor the answer?
To be clear: the definite integral is the difference between the indefinite integral evaluated at the upper and lower limits … here we could write it as I(2) – I(0). The constant “k” is not needed as it’s the same in both cases.
Your turn …
10 2
1Question: Evaluate the definite integral ( 3 4)x x dx
Answer:
103 212 1
3 212
12
12
4
( 1000 50 40) ( ( 1) ( 1) 4( 1))
( 990) (1 4)
995
x x x
Be careful!
The area is negative?! Is it a mistake …?
-350
-300
-250
-200
-150
-100
-50
0
-1-0.7-0.4-0.1 0.
20.50.81.11.41.7 2
2.32.62.93.23.53.84.14.44.7 5
5.35.65.96.26.56.87.17.47.7 8
8.38.68.99.29.59.8
No mistake! The area is just a sum of function values. Itcan be any real number, positive, negative or zero. So be careful when interpreting the result!
You have just calculated the blue area.
23 4y x x
Infinite and improper integrals
Let’s consider difficult cases like:
1
21
1
0
1
1
1
dxx
dxx
dxx
We don’t have a formula forthis (n+1=0), so we’ll ignore itfor now (actually )1
ln | |dx x kx
Improper integrals are integrals over an interval includinga value of x where the function is undefined.Infinite integrals are those involving at least one limit at±∞
This has a finite value
This also has a finite value
Calculate please
A
B
C
Infinite and improper integrals continued
In order to better understand the difference between Aand B, let’s take a close look at the graph:
0
1
2
3
4
5
6
7
8
9
10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
1/x
1/x2
1/rootx
The area under between 1 and ∞ is finite but unbounded.
2
1
x
Infinite and improper integrals continued
The problem of infinite integrals is of course similar to theproblem for infinite sums:
2
21
1
6n n
but1
1
n
… we say it “diverges”
Example
Evaluate:83
1 2 1
xdx
x
523 3
1
2 53 3
33
5
1 3 13 0
51 1
12
5
x x
Notes on Integration
Integration by substitution – ONLY use this for examples where the “inner” function is linear. Other examples are too advanced for P1.
2.5 3.52.5
3 33
3.5dx x dx x
x
Remember the “n+1 over n+1” rule and use it correctly!
1.52x
•Indefinite integrals – you MUST use +k.32217 2
2 3(17 )x x dx x x k !
4(10 2)x dx n = -4. a = 10, so divide by (-3x10)
31(10 2)
30x k
Notes on Integration
Areas found by definite integration are positive if..… the function values are positive (and vice versa)
Areas of regions bounded by two functions can be calculated… by subtracting one function from the other
and then integrating. Result will be positive if … you subtract the “lower” function from the higher one.
( 4)b
ax x dx
Is the integral (area) positive for:a=8 b=10a=-10 b=-8a=1 b=3a=1 b=7 ?