Upload
-
View
51
Download
1
Embed Size (px)
Citation preview
Mathematics Part 2 Mrs. Kristine Sevcenko
Improper Integrals
Recall the definition of a definite integral: . This definition
works only for functions continuous on the segment . If the function has Type II
discontinuity ( ) or if a or b (or both) is infinite then summation is
impossible. In these occasions we have improper integrals.
An improper integral is said to converge if it has a finite value; otherwise it is said to diverge.
Type 1 Improper IntegralsThe function is continuous but one or both integration boundaries are infinite:
(a) ; calculated ;
(b) ; calculated ;
(c) ; his integral is split into two (a) and (b) integrals: where c is any
real number; the integral converges if and only if both components converge.
If is an antiderivative of and a finite limit exists then the integral converges
and .
Type 2 Improper IntegralsThe function has Type II discontinuity at one or both ends of the segment of integration.
(a) function is continuous on but ; calculated
;
(b) continuous on but ; calculated ;
(c) similar situation at both ends of the interval; then it is split into a sum of two integrals
where and converges if and only if both components converge.
Page 1
Mathematics Part 2 Mrs. Kristine Sevcenko
If , is an antiderivative of and a finite limit exists then the integral
converges and .
Similarly, if , is an antiderivative of and a finite limit exists then the
integral converges and .
Type 3 Improper IntegralsThey contain both Type II discontinuities and infinite integration boundaries; these integrals can be split into a sum of finite number Type 1 and Type 2 integrals.
Examples. Find whether the given improper integral converges or diverges; if it converges, evaluate it.
1. 2. 3.
1. Type 1; therefore
and the integral diverges.2. Type 1; split into a sum of two integrals:
3. Type 2: ; diverges.
Exercises . Find whether the given improper integral converges or diverges; if it converges, evaluate it.
1. 7. (a is a constant)
2. 8.
3. 9.
4. 10.
5. 11.
6. 12.
Page 2
Mathematics Part 2 Mrs. Kristine Sevcenko
Page 3