Improper Integrals: Real-Life Applications & Syllabus Impact

[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.

 

[pasmith]

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.

The definition of the Laplace transform is by the improper integral
$$F(s) = \int_0^\infty e^{-st} f(t)\,dt.$$

The definition of the Fourier transform is by the improper integral
$$F(\omega) = \int_{-\infty}^{\infty} e^{i\omega t} f(t)\,dt.$$

The cumulative distribution function of the standard Normal distribution is defined by the improper integral
$$\Phi(z) = \int_{-\infty}^z \frac{1}{\sqrt{2\pi}} e^{-\frac12 z^2}\,dz$$

The space of wavefunctions in quantum mechanics has as its inner product the improper integral
$$\langle f,g \rangle = \int_{-\infty}^{\infty} f(x)g^{*}(x)\,dx$$

Do you need further examples?