Basics of Integration and Derivatives

Roll # 2011-AE-14
Faisal Waqar(14)
Bsc.Building &Architectural
Engineering UCE&T BZU Multan Pakistan.

Basics of
Derivatives
&
Integration

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

A FUNCTION F IS AN ANTIDERIVATIVE
OF A FUNCTION F IF F’=F.
Defination:-

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 

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 

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

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
• 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

Integration is opposite
/ Inverse of
Derivative?????
  

First we learn some
techniques and
theorems on
integration,then

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

BAISIC INTEGRATION FORMULAS
1
1
1. ( 1) 2. ln | |
1
3. 4.
ln

  

 
 
 
n
n
x
x x x
x
x dx n dx x
n x
a
e dx e a dx
a

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
  
  
  
   
 
 
 
 

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

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 

Integration Using Trigonometric
Substitution

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
  
  
  
   
   
   

Integration by partS
( ) '( ) ( ) ( ) ( ) '( )f x g x dx f x g x g x f x dx
udv uv vdu
 
 
 
 

>>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

Definite
IntegralZ
Lets discuss

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..

Rules for Definite Integrals
1) Order of Integration:
( ) ( )
a b
b a
f x dx f x dx  

2) Zero: ( ) 0
a
a
f x dx 

3) Constant Multiple:
( ) ( )
( ) ( )
b b
a a
b b
a a
kf x dx k f x dx
f x dx f x dx

  
 
 

4) Sum and Difference:
( ( ) ( )) ( ) ( )
b b b
a a a
f x g x dx f x dx g x dx    

5) Additivity:
( ) ( ) ( )
b c c
a b a
f x dx f x dx f x dx   

Finally we discuss AREA
UNDER CURVE.

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

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

>>>>>>>>>>>>>>>>>>>>>>>>>
Now see integration
is opposite to
derivative

Applications of
integration related
to course outline..

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

Lengths of curves
Rules of Pappus

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

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

Dividing the whole plane figure up into a number of strips, each will
contribute its own flat disc with volume V   y2.x

The total volume will then be:
As x → 0 the sum becomes the integral giving:
2
x b
x a
V y x 


 
2
x b
x a
V y dx


 

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

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

The total volume will then be:
As x → 0 the sum becomes the integral giving:
2 .
x b
x a
V xy x 


 
2 .
x b
x a
V xy dx


 

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.
. .
. .
2
x b
x a
x b
x a
Ax x y x
y
Ay y x











In the limit as the width of the strips approach zero the sums are
converted into integrals giving:
2
and
1
2
b
x a
b
x a
b
x a
b
x a
xydx
x
ydx
y dx
y
ydx











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


 



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:
2 2 2
2
( ) ( ) ( )
so
1
s x y
s y
x x
  
 
 
 
 
   
 

Lengths of curves
In the limit as the arc length s approaches zero:
and so:
2
1
ds dy
dx dx
 
   
 
2
1
b
x a
b
x a
ds
s dx
dx
dy
dx
dx



 
   
 



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:
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
  
  

  

   
     
       
     
       
          
       


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:
2 . so 2 .
A s
A y s y
x x
 
   
 
 

From previous work:
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



 
   
 
 
   
 
 
   
 


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:
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
 
 
   
    
   


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

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

Basics of Integration and Derivatives

Basics of Integration and Derivatives

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