  We describe an elementary method for bounding a one-dimensional oscillatory
integral in terms of an associated non-oscillatory integral. The bounds
obtained are efficient in an appropriate sense and behave well under
perturbations of the phase. As a consequence, for an $n$-dimensional
oscillatory integral with a critical point at the origin, we may apply the
one-dimensional estimates in the radial direction and then integrate the
result, thereby obtaining natural bounds for the $n$-dimensional oscillatory
integral in terms of the measures of the sublevel sets associated with the
phase. To illustrate, we provide several classes of examples, including
situations where the phase function has a critical point at which it vanishes
to infinite order.
