Practical use and applications of improper integrals Whenever I think of improper integrals, and their applications my mind remembers all the physics based equations Given position at all points, we can use an indefinite integral to find the speed and acceleration at all points We can apply this rule of rates to pretty much anything to find out total work, or total volume of anything
When is the improper integral well-defined in multiple dimensions? Hartman and Mikusinski's book "The Theory of Lebesgue Measure and Integration" make an interesting remark on improper integrals in multiple dimensions: In the case of one variable, we introduced, besides the concept of the Lebesgue integral on an infinite interval, the further concept of an improper integral
Dirichlets test for convergence of improper integrals Improper integrals can be defined as limits of Riemann integrals: all you need is local integrability However, we know that continuity is "almost necessary" to integrate in the sense of Riemann, so teachers do not worry too much about the minimal assumptions under which the theory can be taught