We analyze the error introduced by approximately calculating the
s
s
-dimensional Lebesgue measure of a Jordan-measurable subset of
I
s
=
[
0
,
1
)
s
I^s=[0,1)^s
. We give an upper bound for the error of a method using a
(
t
,
m
,
s
)
(t,m,s)
-net, which is a set with a very regular distribution behavior. When the subset of
I
s
I^s
is defined by some function of bounded variation on
I
¯
s
−
1
{\bar I}^{s-1}
, the error is estimated by means of the variation of the function and the discrepancy of the point set which is used. A sharper error bound is established when a
(
t
,
m
,
s
)
(t,m,s)
-net is used. Finally a lower bound of the error is given, for a method using a
(
0
,
m
,
s
)
(0,m,s)
-net. The special case of the 2-dimensional Hammersley point set is discussed.