Observations about Error in Integral Approximations

The Simplest Geometry

Errors in Left and Right SumsRectangle 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 ApproximationRectangle 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.

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.

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.

A More Precise Way of Seeing This

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.

The box approximation on the left is the same as the trapezoid approximation on the right!

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.