Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
5. When the derivative of function
is given,then the aim to function
itself can be achieved, The
method to find such a function
involves the inverse process of
DERIVATIVE.It is also called
anti-derivation or
INTEGRATION
6. A FUNCTION F IS AN ANTIDERIVATIVE
OF A FUNCTION F IF F’=F.
Defination:-
7. Representation of
Antiderivatives
• If F is an antiderivative of f on an
interval I, then G is an antiderivative
of f on the interval I if and only if G
is of the form , for all
x in I where C is a constant.
G x F x C
8. Some terms to be
familiar with…
• The constant C is called the constant
of integration.
• The family of functions represented
by G is the general antiderivative of
f.
• is the general solution
of the differential equation
• example of general quadratic
G x F x C
.G x F x
9. NOTATION FOR
ANTIDERIVATIVES
• When solving a differential equation
of the form , we solve for ,
giving us the equivalent differential
form .
– This means you isolate dy by multiplying
both sides of the equation by dx. It is
easier to see if you write the left side
as instead of
dy
f x
dx
dy f x dx
dy
dy
dx
.y
10. Notation continued…
• The operation of finding all solutions
of this equation is called
antidifferentiation or indefinite
integration and is denoted by an
integral sign . The general solution
is denoted by
{
}
{
Variable of
Integration
Constant of
IntegratInt ionegrand
y f x dx F x C
11. 11
• f(x): function (it must be continuous in [a,b]).
• x: variable of integration
• f(x) dx: integrand
• a, b: boundaries
b
a
dxxf )(
12.1 Notation
16. As we have seen, integration is more
challenging than differentiation.
In finding the derivative of a function, it is obvious
which differentiation formula we should apply.
However, it may not be obvious which technique
we should use to integrate a given function.
TECHNIQUES OF INTEGRATION
18. TABLE OF INTEGRATION FORMULAS
2 2
5. sin cos 6. cos sin
7. sec tan 8. csc cot
9. sec tan sec 10. csc cot csc
11. sec ln sec tan 12. csc ln csc cot
x dx x x dx x
x dx x x dx x
x x dx x x x dx x
x dx x x x dx x x
19. 1 1
2 2 2 2
13. tan ln sec 14. cot ln sin
15. sinh cosh 16. cosh sinh
1
17. tan 18. sin
x dx x x dx x
x dx x x dx x
dx x dx x
x a a a aa x
TABLE OF INTEGRATION FORMULAS
20.
21. Other one is SUBSTITUTION mETHOd
Try to find some function u = g(x) in
the integrand whose differential du = g’(x) dx
also occurs, apart from a constant factor.
For instance, in the integral ,
notice that, if u = x2 – 1, then du = 2x dx.
So, we use the substitution u = x2 – 1
instead of the method of partial fractions.
2
1
x
dx
x
23. When do we use it?
Trigonometric substitution is used when you have problems involving square
roots with 2 terms under the radical.
You’ll make one of the substitutions below depending on what’s inside your
radical.
2 2 2 2
2 2 2 2
2 2 2 2
sin cos 1 sin
tan sec 1 tan
sec tan sec 1
a u u a
a u u a
u a u a
28. >>Slect the function as second
whose integration is know.
>>If integration of both are
known,take polynomial as first.
>>If no is polynimial then take any
as first.
>>If we are given only one
function whose intgn is
BasicruleZ
32. Previously we were discussing
indefinite integralz in which no limits
are given.
Now we discuss definite integralz. In
these integralz upper and lower limits
are given and we are bounded.
Hey..
33. Rules for Definite Integrals
1) Order of Integration:
( ) ( )
a b
b a
f x dx f x dx
41. 21
1
8
ds
v t
dt
31
24
s t t
31
4 4
24
s
2
6
3
s
The area under the curve
2
6
3
We can use anti-derivatives to find the area
under a curve!
0
1
2
3
1 2 3 4
x
5.2 Definite Integrals
42. Area from x=0
to x=1
0
1
2
3
4
1 2
Example:
2
y x Find the area under the curve from x=1 to x=2.
2
2
1
x dx
2
3
1
1
3
x
31 1
2 1
3 3
8 1
3 3
7
3
Area from x=0
to x=2
Area under the curve from x=1 to x=2.
5.2 Definite Integrals
47. Programme 19: Integration applications 2
Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
48. Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
Programme 19: Integration applications 2
49. Volumes of solids of revolution
If a plane figure bounded by the curve y = f (x), the x-axis and the
ordinates x = a and x = b, rotates through a complete revolution
about the x-axis, it will generate a solid symmetrical about Ox
Programme 19: Integration applications 2
50. Volumes of solids of revolution
To find the volume V of the solid of revolution consider a thin strip of
the original plane figure with a volume V y2.x
Programme 19: Integration applications 2
51. Volumes of solids of revolution
Dividing the whole plane figure up into a number of strips, each will
contribute its own flat disc with volume V y2.x
Programme 19: Integration applications 2
52. Volumes of solids of revolution
The total volume will then be:
As x → 0 the sum becomes the integral giving:
Programme 19: Integration applications 2
2
x b
x a
V y x
2
x b
x a
V y dx
53. Volumes of solids of revolution
If a plane figure bounded by the curve y = f (x), the x-axis and the
ordinates x = a and x = b, rotates through a complete revolution
about the y-axis, it will generate a solid symmetrical about Oy
Programme 19: Integration applications 2
54. Volumes of solids of revolution
To find the volume V of the solid of
revolution consider a thin strip of the
original plane figure with a volume:
V area of cross section ×
circumference
=2 xy.x
Programme 19: Integration applications 2
55. Volumes of solids of revolution
The total volume will then be:
As x → 0 the sum becomes the integral giving:
Programme 19: Integration applications 2
2 .
x b
x a
V xy x
2 .
x b
x a
V xy dx
56. Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
Programme 19: Integration applications 2
57. Centroid of a plane figure
The coordinates of the centroid (centre of area) of a plane figure
are obtained by taking the moment of an elementary strip about the
coordinate axes and then summing over all such strips. Each sum
is then approximately equal to the moment of the total area taken
as acting at the centroid.
Programme 19: Integration applications 2
. .
. .
2
x b
x a
x b
x a
Ax x y x
y
Ay y x
58. Centroid of a plane figure
In the limit as the width of the strips approach zero the sums are
converted into integrals giving:
Programme 19: Integration applications 2
2
and
1
2
b
x a
b
x a
b
x a
b
x a
xydx
x
ydx
y dx
y
ydx
59. Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
Programme 19: Integration applications 2
60. Centre of gravity of a solid of revolution
Programme 19: Integration applications 2
The coordinates of the centre of gravity of a solid of revolution are
obtained by taking the moment of an elementary disc about the
coordinate axis and then summing over all such discs. Each sum is
then approximately equal to the moment of the total volume taken
as acting at the centre of gravity. Again, as the disc thickness
approaches zero the sums become integrals:
2
2
and 0
b
x a
b
x a
xy dx
x y
y dx
61. Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
Programme 19: Integration applications 2
62. Lengths of curves
To find the length of the arc of the curve y = f (x) between x = a and
x = b let s be the length of a small element of arc so that:
Programme 19: Integration applications 2
2 2 2
2
( ) ( ) ( )
so
1
s x y
s y
x x
63. Lengths of curves
In the limit as the arc length s approaches zero:
and so:
Programme 19: Integration applications 2
2
1
ds dy
dx dx
2
1
b
x a
b
x a
ds
s dx
dx
dy
dx
dx
64. Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
Programme 19: Integration applications 2
65. Lengths of curves – parametric equations
Instead of changing the variable of the integral as before when the
curve is defined in terms of parametric equations, a special form of
the result can be established which saves a deal of working when it
is used. Let:
Programme 19: Integration applications 2
2
1
2 2 2
2 2 2
2 2 2 2
( ) and ( ). As before ( ) ( ) ( )
so so as 0
and
t
t t
y f t x F t s x y
s x y
t
t t t
ds dx dy dx dy
s dt
dt dt dt dt dt
66. Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
Programme 19: Integration applications 2
67. Surfaces of revolution
When the arc of a curve rotates about a coordinate axis it
generates a surface. The area of a strip of that surface is given by:
Programme 19: Integration applications 2
2 . so 2 .
A s
A y s y
x x
68. Surfaces of revolution
From previous work:
Programme 19: Integration applications 2
2
2
2
1 and so
2 1 giving
2 1
b
x a
ds dy
dx dx
dA dy
y
dx dx
dy
A y dx
dx
69. Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
Programme 19: Integration applications 2
70. Surfaces of revolution – parametric equations
When the curve is defined by the parametric equations x = f () and y
= F() then rotating a small arc s about the x-axis gives a thin band
of area:
Now:
Therefore:
Programme 19: Integration applications 2
2 . and so 2 .
A s
A y s y
2 2
ds dx dy
d d d
2 2
2
b
x a
dx dy
A y d
d d
71. Volumes of solids of revolution
Centroid of a plane figure
Centre of gravity of a solid of revolution
Lengths of curves
Lengths of curves – parametric equations
Surfaces of revolution
Surfaces of revolution – parametric equations
Rules of Pappus
Programme 19: Integration applications 2
72. Rules of Pappus
1 If an arc of a plane curve rotates about an axis in its plane, the
area of the surface generated is equal to the length of the line
multiplied by the distance travelled by its centroid
2 If a plane figure rotates about an axis in its plane, the volume
generated is equal to the area of the figure multiplied by the
distance travelled by its centroid.
notE: The axis of rotation must not cut the rotating arc or plane
figure
Programme 19: Integration applications 2
73. Learning outcomes
Calculate volumes of revolution
Locate the centroid of a plane figure
Locate the centre of gravity of a solid of revolution
Determine the lengths of curves
Determine the lengths of curves given by parametric equations
Calculate surfaces of revolution
Calculate surfaces of revolution using parametric equations
Use the two rules of Pappus
Programme 19: Integration applications 2