Calculus: Analytic Geometry and Calculus, with Vectors

13

Iterated and multiple integrals

13.1 Iterated integrals When we differentiate a function f having
values f(x) and then iterate (or repeat) the process, we obtain functions
having values denoted by f'(x), f"(x), f"(x),. When we iterate
the process of integrating a function f over an interval from a to x, we
need more elaborate notation to express the results in terms of the given
function. To begin operations, we suppose that

(13.11) fi(x) = faxf(t) dt, f2(x) = faxft(t) dt,

fa(x) =

fa"'f2(t)

dt,

To express f2 in terms off, we can avoid snarls of various kinds by replac-

ing t by tl and then x by t to obtain

fl(t) =f f(tl) dti.

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