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.

~ 168.

Calculation by means of successive integrations.

299

Lim (x, +- x))f(Y+, - ()A,, y -) = f (x, y) d X dy
b:=V(x)
-jid xjf(x, y)dy. ")
a = cp (x)
By the second process likewise we find the double integral equal to:
(iC i (x, Y) dx,
a Xz =(pl (y)
when (P (Y) and 1 (y) denote the values that x assumes at the limits
of the domain for the different values of y.
When the boundary of the domain ex. gr. is the ellipse:
a2 + b2 7
we have:
+ Y=+y6-a-: 2 x —:-T
d xjf(x, y)dy =Ydy f(x, y)dx? - f(x, y)dxdy.
-a b -b a
y = - la2 _x x = by/-y
Y
When the boundary of the domain
is an isosceles right angled triangle
_,_ _ _ ABC with x ranging from a to b;
v~~~~~Y /then while for any value of x, y
A/B passes from a to x, so on the other
hand for any value of y, x varies
between the limits y and b, thus we
z_ _- have:
o _
Fig. 14.
b x b b
Jfff(x, Y) dy= ji X ) j f ( y) f yd
a a a y
a rule already applied in ~ 146, VI.
When the boundary of the domain is a rectangle; for each value
of x, y has the same limits a and j; for each value of y, x has the
same limits a and b; and we obtain from the theorem under discussion
the special theorem:
*) It is to be noticed, that this equation assumes the existence of the
definite double integral; and on this hypothesis proves its identity with the value
resulting from the successive integrations of the right side. It is not allowable
to argue conversely from the right side to the left. (~ 171.)