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∫ ∫
∫∫
=
β
α
θ
θ
θ
θ
φ
θ
φ
)
(
)
(
'
)
,
(
)
,
(
g
f
R
rdrd
r
dA
r
where the region R’ in polar coordinates is R’ = {
α
<
θ
<
β
, f(
θ
) < r < g(
θ
)}.
Double integrals in polar coordinates can be entered in the calculator, making
sure that the Jacobian |J| = r is included in the integrand. The following is an
example of a double integral calculated in polar coordinates, shown step-by-
step: