11. Improper and Singular Integrals

Calculus 2



Weiqi Zhou

Improper Integrals

suppose that is continuous on ，and take . If exists，then it is called the improper integral of on ，and denoted by

Example

  • compute



Improper Integrals

suppose that is continuous on ，and take ，if exists，then it is called the improper integral of on ，and denoted by

Improper Integrals

suppose that is continuous on ，and take an arbitrary ，if both and exist，then is called the improper integral of on ，and denoted by

（or ）

Exercises

  • find

  • find

  • find

  • show that diverges if ，and converges if (and find its value)

Sigularities

if ，then is a singular point of

(strictly speaking only unboundedness is needed here
e.g., at )

Singular Integrals

suppose that is continuous on ，and is a singular point of . If exists，then it is called the singular integral of on , and denoted (still) by

Example

  • find



Singular Integrals

suppose that is continuous on ，and is a singular point of . If exists，then it is called the singular integral of on, and denoted by

Singular Integrals

suppose that is continuous on ，and is a singular point of . If both and exist，then is called the singular integral of on ，and denoted by

Exercises

  • find singular points of following functions



Exercises

  • compute following integrals





Exercises

  • decide whether following integrals are convergent or not





Pay Particular Attentions

if there are singular points inside the interval of integration

Challenge

why don't we define improper integrals the same way as we define definite integrals？

Challenge

  • consider

  • show that converges

  • insert nodes into ，and let ，verify that as , may depend on the choice of nodes

Summary

  • Improper Integrals

  • Singular Points and Singular Integrals

  • Singular Points inside Intervals

  • Mixed Cases