• Improper Integration First & Second Kind CALCULAS(211000)• Civil Department

Prepared by :- Hitesh Maru -(151080106011)

Prashant Nandpal -(151080106012) Anand Ojha -(151080106013) Panchal Sunil -(151080106014)

GUIDED BY:- HEENA PARAJAPATI

Introduction Historically , the concept of definite integral came first as a

means of evaluating area under a curve and clearly this was done to satisfy a geometrical need.

The first idea came from Leibnitz but the first regorous approach was given by Darboux.

The limit of a sum following an idea of Riemann :-

The problem posed by the limiting operation was overcome by an alternative arithmetical approach by Riemann and the equivalence of the two approaches was established.

Basic Definitions

1) Bounded about set A subset E of R is said to be bounded above if there exist some real number X such that a x , for all aE .That is all the elements of E lie to the left if x , upto x atmost . Any such number x is called an upper bound for the set E .

2) Bounded below set A subset E of R is said to be bounded below if there exist some real number x such that x a , for all aE.

Improper integrals• These are a special kind of limit. An

improper integral is one where either the interval of integration is infinite, or else it includes a singularity of the function being integrated.

Improper integral of first kind•

So we have•

Examples•

Improper integral second kind

• In the definite integral ; f(x) becomes infinite at x=a or x=b or at one or more points within the interval (a , b), then the integral is called improper integral of second kind or improper integral with unbounded integrand.

Hence we have•

Example•

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