74 
IMPROPER MULTIPLE INTEGRALS 
Now for any n 
Ca< C fa< fa 
—2Í« S'» J Z n J Kn 
Hence ^f=iim f Ca = a um ft 
n— oo «/93 n=» c/58 
or 
a = lim fg„. 
í¡,=oo «£53 
(15 
Now for a fixed n, 7 may be taken so large that for all points 
of 53, 
e v >e„. 
Hence 
Hence 
Hence 
21 
(5 > lim (L > (L. 
= fe> fr> r<s„ = H. 
*7» JjQ JjQ 
21= f r, 
¡733 
and thus r is integrable in 53. 
This result in 14) gives, on using 58, 3, 
f fff=f (*/+№ 
Jw, J(S J¿8 ./(£ 
(16 
(17 
From 12), 13), and 17) follows 1). 
77. As corollaries of the last theorem we have, supposing 21 to 
be as in 76, 
1. Iff is integrable in 21 andf> — (7, then 
f /= f ff- 
Iff <0-, then r(' r j 
J33 Je 
2. Iff>—G- and I is divergent, ¿Aen 
•Alf 
/// 
are divergent