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.

300 General theorems concerning the double integral. Bk. III. ch. VIII.
When f(x, y) is an integrable /fmetion, we have:
b j3 ft b
j dxjficx, y) (y = dy ff(x, y)dx.
a a oa a
This theorem contains an extension of the condition under which the
order of integrations can be interchanged in the integration of a function
with respect to a parameter, as it was given in ~ 153. For this only
requires as a sufficient condition, that the function f(x, y) should be
finite and doubly integrable, and no longer its complete continuity in
a domain.
169. A double integral having independent limits:
b 1l
f dxjf(x, y)dy
a cs
when regarded as a function of its upper limits ( (b, P) is a continuous
function of both these quantities. For, on the hypothesis that b and
/3 are within the domain of integration of f, we have:
b +7h f+k b ft
0 (b + ht, P + 7k) - (b;, P) =jdx f(x, y)d y - fdxf(x, ) dy
a a a a
b ~i.+ b+~ la +k
=/Jdx J, y) dy + J dxJ f(x, y)dy
a tP b a
+ kc b b-h +4k
=jdy ff(x, y)dx + jdx j f(x, y)dy,
/y a b a
since the order of integrations may be interchanged. This equation
leads to the form:
O (b +, 3 + 7t) - 0(b, /3) = kM + hN,
in which 3I and N are finite quantities; the condition of continuity
therefore is fulfilled.
Moreover the partial derived functions with respect to b and /,
are found from this equation. Putting k- = 0 we find:
(b + i, ) _ (b, ) =fJ f(x, y) dy.
b a
If then Jf(x, y)dy is a continuous function of x in the interval
a
from x - b to x = b + h and we denote a mean value of it by:
a ff(+,t)dy},
a x —b —h