MHB Real World Apps of Improper Integrals: Impact & Syllabus

[matqkks]

What are the real life applications of improper integrals? Why are they on the syllabus of every first course in calculus?
I am looking for examples which have a real impact.

 

[Jameson]

What are the real life applications of improper integrals? Why are they on the syllabus of every first course in calculus?
I am looking for examples which have a real impact.

My area of interest is primarily statistics and although we don't do these by hand at all, improper integrals are used all the time when calculating probabilities on the normal curve. In hypothesis testing we often ask what is the probability something is at least some value... this in essence is asking to take an integral from $[a,\infty]$.