Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 27



Warm-up-exercises

Let

 

be an increasing function and  . Show that the sequence  ,  , converges to   if and only if

 

holds, i.e. if the limit of the function for   is  .



Let   be an interval,   a boundary point of   and

 

a continuous function. Prove that the existence of the improper integral

 

does not depend on the choice of the starting point  .



Let

  be a bounded open interval and

 

a continuous function, which can be extended continuously to  . Prove that the improper integral

 

exists.



Formulate and prove computation rules for improper integrals (analogous to

the rules of definite integrals.



Decide whether the improper integral

 

exists.



Determine the improper integral

 



Let   be a bounded interval and let

 

be a continuous function. Let   be a decreasing sequence in   with limit   and let   be an increasing sequence in   with limit  . Assume that the improper integral   exists. Prove that the sequence

 

converges to this improper integral.





Hand-in-exercises

Compute the energy that would be necessary to move the Earth, starting from the current position relative to the Sun, infinitely far away from the Sun.



Decide whether the improper integral

 

exists and compute it in case of existence.



Give an example of a not bounded, continuous function

 

such that the improper integral   exists.



Decide whether the improper integral

 

exists and compute it in case of existence.



Decide whether the improper integral

 

exists.



Decide whether the improper integral

 

exists.


(Do not try to find an antiderivative for the integrand.)