Observations about Error in
Integral Approximations
The Simplest Geometry
Errors in Left and Right Sums
Rectangle by rectangle, left
Riemann sums underestimate the
area under the graph (since they
always “sample” the smallest value
of the function on the interval). . .
. . .whereas right Riemann sums
overestimate it.
Note that the reverse is true for decreasing functions.
Errors in Trapezoidal Approximation
Rectangle by rectangle, the
trapezoidal approximation
underestimates the area under
the graph of functions that are
concave up. . .
. . . and overestimates the area
under the graph of functions that
are concave down.
Errors in Midpoint Approximation
It may be a bit less
obvious, but midpoint
Riemann sums routinely
(rectangle by rectangle)
underestimate functions
that are concave up and
overestimate functions
that are concave down.
The portion of the graph
where the midpoint sum
overestimates is shown
in green, where the part
of the graph where the
midpoint sum
underestimates is shown
in red.
Notice that in the concave up graph
of the left, the red region is larger in
area than the green region. In the
concave down graph on the right, the
reverse is true.
A More Precise Way of Seeing This
The box
approximation
on the left is the
same as the
trapezoid
approximation
on the right!
But this fact has nothing to do with tangent lines.
Every trapezoid (with parallel vertical sides) has
area given by the width times its height at the
midpoint. We just picked the one whose top side is
tangent to the curve.
Trapping the Value of the Integral
If you have a monotonic function, then the
actual value of the integral is “trapped”
between the left and right Riemann sums.
They provide an upper and lower estimate
for the integral.
If you have a function that is purely
concave up or concave down, the
midpoint and trapezoidal approximations
provide an upper and lower estimate for
the actual value of the integral.