An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

270 The definite integral as the limiting value of a sum. Bk. III. ch. VI.
a within certain limits /s and y, then it is also between these limits
undoubtedly integrable with respect to a.
b
Calling: ff(x, a) dx F= F(a), we have:
C6
a y b
F(a)da ==dfa fx, a)dx.
p/3t byt a
To such an expression the name definite double integral has
been given; it implies, that first the integration regarding x has to
be effected and then that regarding a. The general theory of double
integrals will be treated in Chapter VIII; one question only is to be
solved here. Assuming the limits a and b, jt and y, to be independent
determinate constants, and f(x, a) to be a continuous function of both
variables, does the following equality hold between the
integrals:
y b b Y
da Jf(x, a) x = dx f(x, a)da;
If a a
or, does the order in which we integrate influence the result?
For b = a both expressions vanish. They are therefore equal functions
of b, if their derivates with respect to b coincide. Differentiating each
with respect to the upper limit b, we find on equating:
y b Y7 Y
abfJd aj X, a)dclx= J fJ(x, a) d a jf(b, a)da.
tl a f forx=ab t
Now this requires the integral with respect to ac on the left side to
admit of differentiation under the integral sign. But it does admit
of it, because the derivate of
b
f f(x, a)dx
a
with respect to b, whose value is f(b, a), is by hypothesis a continuous
function of both a and b. The order of the integrations can therefore be
inverted for a continuous function of two variables.
When the limits b and y are infinite, we have:
daj f (x, a) dx = Linm da a Lim _ f(xa) dx
Co CI U IVdx f(x, a) dl = Lirn dx { Lim f(x, a) da.
aix:~~~~~~~(" -- C O
a ~ ~ ~ ~ 1 =c 0=c