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# Kalkulus II (3 - 4)

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### Kalkulus II (3 - 4)

1. 1. Kalkulus II<br />Teguh Budi P, M.Si <br />Sesion#03-04<br />JurusanFisika<br />FakultasMatematikadanIlmuPengetahuanAlam<br />1/9/2011<br />1<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
2. 2. Outline<br />Improper Integrals<br />1/9/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />2<br />
3. 3. Improper Integrals<br />1/9/2011<br />3<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
4. 4. Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals.<br />Until now we have been finding integrals of continuous functions over closed intervals.<br />1/9/2011<br />4<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
5. 5. Example 1:<br />The function is undefined at x = 1 .<br />Can we find the area under an infinitely high curve?<br />Since x = 1 is an asymptote, the function has no maximum.<br />We could define this integral as:<br />(left hand limit)<br />We must approach the limit from inside the interval.<br />1/9/2011<br />5<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
6. 6. Rationalize the numerator.<br />1/9/2011<br />6<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
7. 7. This integral converges because it approaches a solution.<br />1/9/2011<br />7<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
8. 8. Example 2:<br />(right hand limit)<br />We approach the limit from inside the interval.<br />This integral diverges.<br />1/9/2011<br />8<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
9. 9. The function approaches<br />when .<br />Example 3:<br />1/9/2011<br />9<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
10. 10. 1/9/2011<br />10<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
11. 11. Example 4:<br />(P is a constant.)<br />1/9/2011<br />11<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
12. 12. Thank You<br />1/9/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />12<br />