Quick description
This is an utterly trivial bound: one can bound the magnitude of an integral by
, where the "base"
is an upper bound for the measure
of
, and
is an upper bound for the magnitude
of
on
. Thus for instance

If is a signed or complex measure, one must ensure that
bounds the total variation
and not just
. (Mistakes have been made because this issue was neglected!)
One can also get lower bounds this way: if is unsigned and
is bounded from below by
, and
is bounded from below by
, then
is bounded from below by
. Note however that it is not enough to have
bounded from below by
to draw a non-trivial conclusion; one must lower bound
itself.
Prerequisites
Basic measure theory; calculus; complex analysis
Example 1
Suppose one wants to show that the contour integral

goes to zero as , where
is the upper semicircle
. We can use "base times height" here. For the "base", observe that the length
of
(which is also the measure of
with respect to
) is just
. For the "height", observe that on the upper semicircle,
has magnitude at most
, while
is bounded from below by
(for
), and so
is bounded from above by
, thus leading to a total height bound of
. This leads to a total bound for the magnitude of integral of
, which goes to zero as
, and the claim follows from the squeeze test.
(Note that this technique does not work if the denominator was linear instead of quadratic. To see how to deal with that case, see the article "bound by a Riemann sum".)
General discussion
The base times height bound is reasonably efficient as long as
-
does not oscillate significantly (i.e. there is no significant opportunity for cancellation); and
-
the magnitude of
is reasonably uniformly distributed across
(no "spikes" or large "empty regions").
When the latter condition fails, then it is often advantageous to first subdivide the integral into regions (i.e. divide and conquer) where the magnitude of the integrand is reasonably uniform, so that this method can be applied. This leads to such methods as "bound by a Riemann sum" and "dyadic decomposition".
The base times height method can also be used as a heuristic to guess the right size of an integral. If a function f is "expected" to "concentrate" on a subset of measure about , and is also believed to typically have a value of about
on this subset, then it is reasonable to expect that the final integral is of order
(unless one expects lots of cancellation in the integral, in which case it could have a considerably smaller magnitude). For instance, consider the integral
for some parameter
. Graphing this function, we see that it attains a "height" of about
and is concentrated on a "base" interval of length about
, so a reasonable first guess for this integral is that it should be about
, which is indeed the case (see the article "bound the integrand by something simpler" for more discussion).
The following variant of the base times height bound is also useful: if a function has height at most
almost everywhere on a set
of measure at most
, then the
of
for any
is at most
.