Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

270 Multiple and Line Integrals [Ch. 7
2. Improper double and triple integrals.
a) An infinite region. If a function f (x, y) is continuous in an unbounded
region 5, then we put
\{f(x, y) dx dy = lim (
\ f (x, y) dx dy, (1)
" S U
W
where a is a finite region lying entirely within S, where a -+ S signifies that
we expand the region o by an arbitrary law so that any point of 5 should
enter it and remain in it. If there is a limit on the right and if it does not
depend on the choice of the region o , then the corresponding improper integral
is called convergent , otherwise it is divergent.
If the inU'grand / (,v, //) is nonnegative [f (x, y)^Q], then for the convergence
of an inirioper integral it is necej-sary and sufficient for the limit
on the right of (1) lo exist at least for one system of regions o that exhaust
the region 5.
b) A discontinuous function. If a function / (x, y) is everywhere continuous
in a bounded closed region S, except the point P (a, b), then we put
(*, y)dxdy=\\m f(x,y)dxdy. (2)
where S6 is a region obtained from S by eliminating a small region of dia
meter e that contains the roint P. If (2) has a limit that does not depend
on the tyre of small regions eliminated from 5, the improper integral under
consideration is called convergent, otherwise it is divergent.
If /(A, //)^>0, then the limit on the ripht of (2) is not dependent on the
type of regions eliminated from S; for instance, such regions may be circles
of radius with centre at P.
The concept of improper double integrals is readily extended to the case
of triple integrals.
Example 2. Test for convergence
is}
dxdy
where S is the entire j/-plane.
Solution. Let a be a circle of radius Q with centre at the coordinate
origin. Passing to polar coordinates for p^ 1, we have
If p< 1, then lim 7 (a) = lim /(a) =00 and the integral diverges. But if p> 1,
O -* S ~f Q QC
then lim 7(o)= r and the integral converges. For p=l we have
o - a P *
,
3)