In this paper we consider the problem of computing 2-D Cauchy principal value integrals of the form
\[
₯
S
F
(
P
0
;
P
)
d
P
,
P
0
∈
S
,
{\fint _S}F({P_0};P)\,dP,\qquad {P_0} \in S,
\]
where S is either a rectangle or a triangle, and
F
(
P
0
;
P
)
F({P_0};P)
is integrable over S, except at the point
P
0
{P_0}
where it has a second-order pole. Using polar coordinates, the integral is first reduced to the form
\[
∫
θ
1
θ
2
[
u
n
k
]
0
R
(
θ
)
f
(
r
,
θ
)
r
d
r
d
θ
,
\int _{{\theta _1}}^{{\theta _2}} {[unk]_0^{R(\theta )}\frac {{f(r,\theta )}}{r}dr\, d \theta ,}
\]
where
[
u
n
k
]
[unk]
denotes the finite part of the (divergent) integral. Then ad hoc products of one-dimensional quadrature rules of Gaussian type are constructed, and corresponding convergence results derived. Some numerical tests are also presented.