The layer cake representation follows easily from observing that
and then using the formula
The layer cake representation takes its name from the representation of the value as the sum of contributions from the "layers" : "layers"/values below contribute to the integral, while values above do not.
It is a generalization of Cavalieri's principle and is also known under this name.[1]: cor. 2.2.34
An important consequence of the layer cake representation is the identity
^Willem, Michel (2013). Functional analysis : fundamentals and applications. New York. ISBN 978-1-4614-7003-8.{{cite book}}: CS1 maint: location missing publisher (link)
Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.