Calculus: Analytic Geometry and Calculus, with Vectors

13.1 Iterated integrals 653

Substituting this in the formula for f2(x) then gives

(13.12) .12(x) = fax (fatf(ti) dti) dt.

Replacing ti by t2 and then t by ti and x by t gives

f2 (t) =f t ( at`a f(t2) dtZ) dti

and substituting this into the formula for f3(x) gives

(13.13)

f3(x) = fax l ft (fats f(ts)

dt2) dtl} dt.

The integrals in (13.12) and (13.13) are examples of iterated integrals

and, for each n, we could write a formula for ff(x) which involvesn of

these integrals.

In case f(x) = 1 for each x, we do not need the iterated integralsto

obtain formulas for fi(x), f2(x), ; we can use the formulas (13.11)

one after another to obtain

(13.14) fi(x) = x - a, f2(x) _ (x - a)f3(x) _(x - a)

2! 3!

f4(x) _ (x

41

a)4

We must, however, learn how iterated integrals are manipulated when

they appear in our work and cannot be avoided. The way in which

(13.12) was obtained tells us that to get f2(x) we should integrate first

with respect to ti to evaluate the integral in parentheses to obtain a

function of t which is integrated with respect to t to obtain f2(x). A

tempest in a teapot appears when we, like everyone else, adopt the view

that parentheses are nuisances and write (13.12) in the form

(13.15) f2(x) = f ax ftf(ti) dti dt

and insist that we must find f2(x) by integrating first with respect to ti,

the limits of integration being a and t, and integrating last with respect

to t from a to x. The difficulty lies in the fact that there is always the

possibility of constructing a theory of iterated integrals in such a way

that, for example,

(13.16) f ' f D F(x,y) dx dy

means the result of integrating first with respect to y from C to D (not

from A to B) and then integrating last with respect to x from A to B.

It is equally sensible to insist on one hand that we should "work outward

from the middle," so that fc goes with dx and f4 goes with dy, and toD