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improper integrals
- 1. •Improper Integration First& Second Kind CALCULAS(211000) •Civil Department
- 2. Prepared by:- Hitesh Maru -(151080106011) Prashant Nandpal -(151080106012) Anand Ojha -(151080106013) Panchal Sunil -(151080106014) GUIDED BY:- HEENA PARAJAPATI
- 3. 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.
- 4. Basic Definitions 1) Boundedabout 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 a∈E .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 a∈E.
- 5. Improper integrals • Theseare 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.
- 6. Improper integral offirst kind •
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- 9. Improper integral secondkind • In the definite integral 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 ; 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.
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