Is a Midpoint Riemann Sum an Over or Underestimate?

Understanding whether a midpoint Riemann sum over or underestimates the true area under a curve is crucial in numerical analysis and calculus. The accuracy of this approximation depends on the concavity of the function being integrated over an interval.

Key Points to Consider

Concave Up Functions: If the function is concave up, meaning the second derivative is positive over the interval, the midpoint Riemann sum will be an overestimate. Concave Down Functions: If the function is concave down, meaning the second derivative is negative over the interval, the midpoint Riemann sum will be an underestimate. Linear Functions: For linear functions, the midpoint Riemann sum will exactly match the area under the curve as there are no curves to consider.

General Rules and Analysis

The behavior of a function with respect to its concavity can be summarized as follows:

For a continuous function (f(x)) on the interval ([a, b]), if (f''(x) Conversely, if (f''(x) > 0) (concave up) over the interval, the midpoint Riemann sum (M_n) is an underestimate.

This can be observed by examining the function over a subinterval ([x_{i-1}, x_i]). Let (x_i^*) be the midpoint of this subinterval. The midpoint Riemann sum contribution (f(x_i^*) Delta x) represents the area below the horizontal line (y f(x_i^*)) over ([x_{i-1}, x_i]). This area is unchanged if the line is replaced by the tangent line to (y f(x)) at (x_i^*).

If (f(x)) is concave down, the area below the tangent line is less than the area below (f(x)) itself over ([x_{i-1}, x_i]). If (f(x)) is concave up, the area below the tangent line is more than the area below (f(x)) itself over ([x_{i-1}, x_i]).

Additional Approximations and Insights

Comparison with Trapezoidal Rule

For a function (f(x)) that is concave down across the entire interval ([a, b]), the trapezoidal approximation (T_n) for (int_{a}^{b} f(x) ,dx) will be an underestimate. On the other hand, if (f(x)) is concave up across ([a, b]), the trapezoidal approximation (T_n) is an overestimate.

Observing that even for functions that are neither strictly concave up nor strictly concave down, the midpoint approximation (M_n) and the trapezoidal approximation (T_n) tend to offset each other. If one is too large, the other is too small. More precisely, the midpoint sum (M_n) tends to be off by only about half as much as the trapezoidal sum (T_n).

Improving the Approximation: Simpson's Rule

Using (frac{2}{3} M_n frac{1}{3} T_n) as an estimate can help improve the approximation, though it requires about twice as much computational effort compared to either (M_n) or (T_n) alone. For practical purposes, one might choose an even (n) and use (S_n frac{2}{3} M_{n/2} frac{1}{3} T_{n/2}) as the approximation. This is exactly what is known as Simpson's rule.

Simpson's Rule in Practice

Simpson's rule is usually introduced by approximating (f(x)) with second degree interpolating polynomials over two subintervals at a time. However, this approach can be less engaging. In my experience, a more intuitive and engaging way to introduce Simpson's rule involves directly observing the function and its behavior over the interval. This method not only aids in understanding but also makes the process more enjoyable for the students.