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.)