# Spherical coordinates to compute a definite integral

The introduction of polar coordinates is often motivated (among others) with the possibility of calculating definite integrals of functions with non-elementary antiderivatives. This is the case with the Gaussian integral $\int_{-\infty}^{\infty}e^{-x^2}dx$ which can be easily calculated by computing the corresponding double integral in two ways (one using Fubini's theorem and the other by using polar coordinates).

I am looking for a similar integral that would motivate the introduction of spherical coordinates.

More specifically what I want is

-A definite integral such that the antiderivative is non-elementary.

-For pedagogical reasons it is preferable that the integrand is a quite elementary function

-Can be calculated by using the spherical coordinates and a triple integral (perhaps in the same way as the Gaussian integral)

-Can not be calculated by more elementary methods, specifically the method using polar coordinates fails.

• "The introduction of polar coordinates is often motivated (among others) with the possibility of calculating definite integrals of functions with non-elementary antiderivatives". I disagree with you, I think that solving these integrals is nice feature but far from being the motivation for introducing polar coordinates. What you're saying applies more to the residue theorem than physical coordinates of symmetry in 2D and 3D space – Ofek Gillon Apr 28 '17 at 10:50
• Of course, from the point of view of somebody who already knows such things as differential equations, mathematical physics and so on, it is evident that polar coordinates have far more important applications than calculating some integrals. Yet somebody who will ask you ,,why do we need to study such stuff'' likely has no knowledge of such fields. So telling him ,,it is motivated by differential equations'' is equivalent to ,,you will learn later''. Showing him the ,,integral'' application leaves him with the feeling that he just learnt a new tricky tool which is not totally useless – hassan Apr 28 '17 at 12:03
• Why introduce polar and spherical coordinates with differential equations? I like to introduce the concept as an easier way to describe space in some cases. – Ofek Gillon Apr 28 '17 at 12:15