Cumulative weighted integral

For a locally integrable weighted source $r^ef(r)$ , its cumulative weighted integral from a fixed $a>0$ is

I_e f(R)=\int_a^R r^e f(r)\,dr.

When integration from zero converges, $a=0$ is also allowed. The upper endpoint $R$ remains a variable, unlike a total radial moment.

Derivatives and constants

If the source is continuous, $\partial_R I_ef(R)=R^ef(R)$ . An arbitrary primitive differs by a constant, which can depend on any additional parameters. Matching sources on an exterior interval therefore does not by itself match their primitives there; the accumulated integral or an integration constant must also agree.