Theory

Introduction

Iterated integrals are defined in other coordinate systems in exactly the same manner. However, they do not give the expected geometric quantity unless extra factors are added to create an infinitesimal unit of area. For polar coordinates, the unit of area is $dA=rdrd\theta$. For spherical coordinates, the unit of volume is $dV=r^2\sin (\phi )drd\theta d\phi$.

Introduction (part 2)

Area measure requires correction factors

What happens if we neglect the factor ‘$r$’ in $dA=rdrd\theta$ and simply compute ${\int }_c^d{\int }_a^{b}f(r,\theta )drd\theta$? We can still perform this iterated integral. But the resulting quantity corresponds to the integral of the function $f(r,\theta )$ over the box $(r,\theta )\in [a,b]\times [c,d]$, as if $r$ and $\theta$ were Cartesian coordinates.