Sampled Sums

Let $f$ be a continuous, real-valued function on some closed interval $[a, b]$. A sampled sum for $f$ on $[a, b]$ is the total signed area of a collection of abutting rectangles whose heights are function values of $f$. See below for definitions.

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Click the diagram to add points to the splitting, or use the field below to create an equal-length splitting. (Clicking on an existing point removes that point from the splitting.)

Intervals = ,
Estimate = ,
Error = .

Definitions

Assume $[a, b]$ is a closed, bounded real interval and $n$ is a positive integer. A splitting of $[a, b]$ is a finite subset $\Pi = \{x_{i}\}_{i=0}^{n}$ of $[a, b]$ that contains both endpoints. Re-indexing if necessary, we may assume \[ a = x_{0} \lt x_{1} \lt \dots \lt x_{n-1} \lt x_{n} = b. \] For each $i$ such that $0 \leq i < n$, call $I_{i} = [x_{i}, x_{i+1}]$ the $i$th piece, and let $\Delta x_{i} = x_{i+1} - x_{i}$ be its length.

If $\Pi$ is a splitting of $[a, b]$, a set $x^{*} = \{x_{i}^{*}\}_{i=0}^{n-1}$ such that $x_{i} \leq x_{i}^{*} \leq x_{i+1}$ for each $i$ is called a set of sample points from $\Pi$. The corresponding sampled sum from $\Pi$ is \[ S(f, \Pi, x^{*}) = \sum_{i=0}^{n-1} f(x_{i}^{*})\, \Delta x_{i}. \]

Geometrically, split $[a, b]$ into pieces. Select a sample point from each piece, evaluate the function $f$ at this point, and construct the rectangle of this height lying over the $i$th piece. The resulting sampled sum is the total signed area of these rectangles; that is, if $f(x_{i}^{*}) \lt 0$, we count the area of that rectangle negatively.

If $x_{i}^{*} = x_{i}$, the left-hand endpoint of $I_{i}$, we call $S(f, \Pi, x^{*}) = \texttt{LEFT}(f, \Pi)$ the left-hand sampled sum.
If $x_{i}^{*} = x_{i+1}$, the right-hand endpoint of $I_{i}$, we call $S(f, \Pi, x^{*}) = \texttt{RIGHT}(f, \Pi)$ the right-hand sampled sum.
If $x_{i}^{*}$ is a point of $I_{i}$ where $f$ achieves its maximum in $I_{i}$, we call $S(f, \Pi, x^{*}) = U(f, \Pi)$ the maximum sampled sum.
If $x_{i}^{*}$ is a point of $I_{i}$ where $f$ achieves its minimum in $I_{i}$, we call $S(f, \Pi, x^{*}) = L(f, \Pi)$ the minimum sampled sum.
If $x_{i}^{*} = \frac{1}{2}(x_{i} + x_{i+1})$, the midpoint of $I_{i}$, we call $S(f, \Pi, x^{*}) = \texttt{MID}(f, \Pi)$ the midpoint sampled sum. The midpoint sum may also be interpreted as the total signed area of trapezoids enclosed by tangent lines to the graph at the midpoints.

One other sum is represented geometrically by the signed areas of trapezoids. The average of the left- and right-hand sums, called the trapezoid sum, is the signed area enclosed by secant lines. That is, draw a dot on the graph of $f$ over each point of the splitting, and connect the dots in order. The trapezoid sum is the resulting signed area.

Relative Accuracy

If $S$ is a sampled sum for $f$ on the interval $[a, b]$, the difference \[ E = S - \int_{a}^{b} f(x)\, dx \] is called the error. An overestimate has positive error; an underestimate has negative error. The smaller the absolute error $|E|$, the more accurate the approximation.

For twice-differentiable functions whose second derivative is continuous, and for equal-length splittings with $n$ pieces, the left-hand, right-hand, maximum, and minimum sums have error roughly proportional to $1/n$. That is, doubling the number of intervals roughly halves the error.

For the same functions and splittings, the trapezoid and midpoint methods have error roughly proportional to $1/n^{2}$. That is, doubling the number of intervals roughly quarters the error. Moreover, the midpoint error is roughly minus one-half the trapezoid error. (The weighted average \[ \texttt{PARA}(f, \Pi) = \tfrac{1}{3}\bigl(\texttt{TRAP}(f, \Pi) + 2\texttt{MID}(f, \Pi)\bigr), \] called the parabolic sum, cancels these second-order errors and is sufficiently accurate for industrial use. This program does not investigate the parabolic sum.)

Qualitative Properties

Try to establish the properties stated below; first convince yourself they are true, then give reasons based on the definitions.

The maximum sum always overestimates, and the minimum sum always underestimates.
If $f$ is increasing on $[a, b]$, then the left-hand sum underestimates and the right-hand sum overestimates. What can you say if $f$ is decreasing?
If $f$ is convex (concave up) on $[a, b]$, then the midpoint sum underestimates and the trapezoid sum overestimates. What can you say if $f$ is concave (down)?