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.

298

General theorems concerning the double integral. Bk. II. ch. VIII.

Proceeding successively to the limits, as the differences (yv+i - yv)
and (x+1 - x,) converge to zero, let us in the first case conceive
the differences (y,+i - yv) chosen so small, that the value of:
2 (y+l -- yv)/'(x,, Yv)
v
shall differ from the value of the definite integral:
ff (x, y) dy
1(A)
only by a quantity A(x,) whose absolute amount can be rendered
arbitrarily small by diminishing the distances yv+ - y,.
For inasmuch as the function f, assumed everywhere finite, is
subject to the conditions of ~ 167, f is generally integrable with
respect to y, and it is only for a discrete set of values of x, that
the integral can lose its meaning. Therefore:
y,( )
-( - yV+ )ff(x, y) - f (xC, y) Cdy + A (xY).
V ~?yo(~)
The quantities yo(') and y1(^) signify the values of y belonging to x,
at the limits of the domain, we can also denote them as functions of
xu by the equations:
y _) - g9 (XZ), Y(It,) =_ (x,).
From the above equation we have further:
y (U)
^(X1- )2(Y+1 - Yv) txl Yv ) fXt =4x — x'y) (ly
+ 2(X+-X) A (x).
Now letting each interval x,,+ - xy, converge to zero, the limiting
values of the quantities upon the right side become:
dxjf'(x, y)dy + A(x) x.
a J) (x) a
But since the absolute amount of A (x) for all values of x, with
possibly a discrete set of exceptions that do not influence the value of
the integral, can be made arbitrarily small, the value of this second
integral is also arbitrarily small, i. e.: