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.

272 The definite integral as the limiting value of a sum. Bk. III. ch. V.

f(X, ac)dx
c-a
smaller than an arbitrarily small quantity,. The integral on the
right side then, as a continuous function of A, also passes over into
the same value, provided it has any determinate limiting value.
These sufficient conditions are fulfilled, for instance, when:
A
abs f(x, a) < - (V <1)
while A as a function of a remains finite.
0
Example.. pin I- dx =_- - er (~~ 150, 152, 155) for every finite
value of a; except for a - 0, where the value of the integral vanishes.
Nevertheless an integration with respect to a is possible ex. gr. between
the limits 0 and 1:
i c Sil a d x a za - clx = -
0 0 0
Interchanging the order of integrations we obtain:
CO 1 CX oD
dx_ i. i;-cos x t (sin X)2
I Isin axda (= x I dx Z.
J d- j. oxcx J1 x2o J x29-d = 2 -0 0 0 0
The following Example shows that an interchange of the order of
integrations in discontinuous functions gives rise to a different value*):
1 1 1 1
t- / (o{2 -., /' n^a2 - X2xda
J *J (o2 - x)c)dx is not equal toJ J (x 2
jd J 2 + X2)J (U2 + X2)2
0 0 0
Here the function to be integrated is discontinuous for x = 0, a= 0.
We have:
1 1
/(Uc2 - X2)dx _ x X _ 1_
J (a2 + x2)2 Vo9 + -2 ~ r+ 2 '
o ~ 0
Hence the first double integral is equal to:
1 1
__- =_ (tan-l a =
0 0
On the other hand:
1 1
J (a2 + x2)9 d 2 -+ X2 I + x42 I
0 0
therefore the second double integral is equal to - 4
a) Cauchy, Le9ons de calcul differentiel et integral, r6dig6es par Moigno. p. 85.