Integral Calculus

1. INTEGRAL CALCULUS
By-
Manish Sahu
M.Sc. Chemistry (Final)
Sp.- Physical Chemistry

2. SYNOPSIS
 Introduction
 History
 Definition
 Rules
 Graphical Representations
 Techniques of Integration
 Application of Integration
 Conclusion
 Reference

3. INTRODUCTION
 The relationship involving the rate of change of two
variables.
 But also needed to know the direct relationship between of
two variables.
 For example, we may know the velocity of an object at a
particular time, but we may went to know the position of the
object at that time.
 To find this direct relationship , we need to use the process
which is opposite to differentiation . This is called
INTEGRATION.

4. HISTORY
 Integration can be traced as far back as
ancient Egypt before 1800 BC.
 Further developed and employed by
ARCHIMEDES and used to calculate areas for
Parabolas .
 Similar methods were independently
developed in in china around the 3rd century
AD by LIU HUI .
 Next major step in INTEGRAL
CALCULUS came in Iraq when the 11th
century mathematician IBN AL- HAYTHAM
( Known as ALHAZEN in Europe ) .

5.  Also formulated independently by ISAAC
NEWTON and GOTTRIED LEIBNIZ in the late of
17th century .
 Acquired a firmer footing with the development
of limit and was given and a suitable foundation by
CAUCHY in the first half of the 19th century .
 INTEGRATION was first rigorously formalized
,using limits, by RIEMANN.
 Other definitions of INTEGRAL, extending
REIMANN’s and LEBESGUE’s approaches, were
proposed.
HISTORY

6. DEFINITION
This article is about the concept of
definite integral in calculus . For the
indefinite integral , see Antiderivatives.

7. RULES
 ∫ sinx dx = - cosx
 ∫ cosx dx = sinx
 ∫ tanx dx = - log(cosx)
 ∫ cotx dx = log(sinx)
 ∫ secx dx = log(secx +tanx )
 ∫ cosecx dx = log(cosec – cotx )
 ∫ x
n
dx = x
n+1
/ n+1
 ∫ 1/x dx = logx
 ∫ 1 dx = x
 ∫ e
x
dx = e
x
 ∫ ax
dx = ax
/ logea

8. GRAPHICAL
REPRESENTATIONS
In mathematics ,an INTEGRAL
assigns numbers to functions in a way that can describe
displacement, area, volume , And other concepts that arise
by combining infinitesimal data. INTEGRATION is one of
the two main operations of calculus , with its inverse
operation, Differentiation , being the other. Given a function f
of a real variables x and an interval [a,b] of the real line ,the
definite INTEGRAL is defined informally as the signed area
of the region in the xy- plane that is bounded by the graph of
f, the x – axis ,
∫a
b
f (x) dx
vertical lines x=a ,and x=b . The area above the x-axis adds to
the total and that below the x-axis subtracts form the total.

9. TECHNIQUES OF
INTEGRATION
Various techniques of integration ;
 Integration by General Rule
 Integration by Parts
 Integration by Substitution

10. Integration by General Rule
∫ xn
dx = xn+1
/ n+1
Example :- ∫ x4
dx
∫ x4
dx = x4+1
⁄ 4+1
= x5
/ 5

11. Integration By Parts :-
∫ f(x).g(x) dx = f(x) ∫g(x)dx - ∫ {d/dx f(x) ∫g(x)dx } dx
Example :- ∫ x.cosx dx
∫ x.cosx dx = x ∫ cosx dx - ∫ {d/dx(x) ∫ cosx dx } dx
∫ x.cosx dx = x ∫ cosx dx - ∫ { 1. sinx } dx
= x ∫ cosx dx - ∫ sinx dx
= xsinx – (- cosx )
= xsinx + cosx

12. Integration by Substitution
Example :- ∫ 4x
3
(x
4
+1) dx
Let u = x4
+1,
Then du = 4x3
dx
∫ 4x
3
(x
4
+1) dx = ∫ u du
= u
2
/2
= ( x
4
+1 )
2
/2

13. APPLICATION OF INTEGRATION
 The PETRONES TOWERS in Kuala Lumpur experience
high forces due to winds. INTEGRATION was used to design
the building for strength .
 The SYDNEY OPERA HOUSE is very unusual design
based on slices out of a ball . Many Differential equation were
solved in the design of this Building .
 Historically ,one of the first uses of INTEGRATION was in
finding the volumes of wine casks .

14. CONCLUSION
 Very important Mathematical tool .
 Used in many fields .
 Important in Business .
 Helps to estimate things like –
 Marginal Cost
 Marginal Revenue
 Profit
 Gross Loss

15. REFERENCE
 “Mathematics For Chemists” By :-
“Bhupendra Singh”
 “A History of the Definite Integral” By :-
“Kallio & Bruce Victor”