Monotonicity Bounds for Riemann Sums

Why Monotonicity Gives Bounds

Left- and right-endpoint sums approximate an integral by adding the signed areas of rectangles. Their accuracy depends in part on where each rectangle samples the function. If the function rises across an interval, the left endpoint gives a height no greater than the values to its right, while the right endpoint gives a height no less. For a falling function, those comparisons reverse.

This gives a useful conclusion even before calculating the integral: monotonicity tells us whether each endpoint sum is a lower or upper bound. The result applies to partitions with unequal as well as equal subinterval widths.

Partitions and Endpoint Sums

Let \(a=x_0<x_1<\cdots<x_n=b\) be a partition of \([a,b]\). The width of subinterval \(i\) is \(\Delta x_i=x_i-x_{i-1}\), which is positive. For a function \(f\), the left-endpoint sum uses the function value at the left endpoint of each subinterval; the right-endpoint sum uses the value at the right endpoint.

Each term is a rectangle's signed area: its height is a function value, and its width is \(\Delta x_i\). The heights can be negative, so the sums and integral represent signed accumulation, not necessarily ordinary geometric area.

The Endpoint Bounds

Assume \(f\) is integrable on \([a,b]\), as is the case, in particular, when \(f\) is continuous there. For an increasing function, every left-endpoint value is no greater than the values across its subinterval, and every right-endpoint value is no less. Consequently, the left sum is a lower bound and the right sum an upper bound.

The inequalities do not require equal widths, and they do not require \(f\) to be positive. A constant function is both nondecreasing and nonincreasing; in that case, both endpoint sums equal the integral.

Why the Bounds Hold

Consider one subinterval \([x_{i-1},x_i]\). If \(f\) is nondecreasing, then throughout that subinterval its values lie between the endpoint values. Since the subinterval width is positive, the constant functions at those endpoint heights give a lower and an upper rectangle. Integrating the comparison on this subinterval bounds its contribution to the integral.

Adding these inequalities for all subintervals proves the result. The sum of the subinterval integrals is the integral over the whole interval, while the sums of the endpoint rectangles are \(L\) and \(R\).

For a nonincreasing function, the values on each subinterval instead satisfy \(f(x_i)\le f(x)\le f(x_{i-1})\). Repeating the same comparison puts the right sum below the integral and the left sum above it. This is the reason the order reverses.

Worked Examples

Checking Monotonicity in Practice

When a function is differentiable on an interval, a useful test is the sign of its derivative throughout that interval. If \(f'(x)\ge0\) throughout, the function is nondecreasing; if \(f'(x)\le0\) throughout, it is nonincreasing. This follows from the Mean Value Theorem: for any \(u<v\), there is a point \(c\) between them such that \(f(v)-f(u)=f'(c)(v-u)\). Since \(v-u>0\), the derivative's sign determines the sign of \(f(v)-f(u)\).

A derivative test is a convenient way to establish monotonicity when it applies, but the endpoint-bound theorem itself requires monotonicity, not differentiability. For example, a graph or a formula may show that a function is nondecreasing even where a derivative is not available.

A Common Mistake

A frequent error is to memorize “left is an underestimate” without checking whether the function is increasing. Consider the decreasing function \(f(x)=4-x\) on \([0,3]\) with partition \(0,1,3\). Its sums are \(L=10\) and \(R=5\), while the integral is \(\frac{15}{2}\). The incorrect claim \(L\le\int_0^3 f(x)\,dx\le R\) would say \(10\le\frac{15}{2}\le5\), which is false.

Using the Bounds

Monotonicity provides a quick check on an approximation and a way to report a guaranteed range when the exact integral is not known. After identifying the direction of change, keep the partition widths positive, form both sums using the appropriate endpoints, and place the integral between them in the correct order.

Remember that increasing or decreasing describes how function values change as \(x\) increases; it does not describe whether the values themselves are positive or negative. The inequalities compare signed numerical quantities, so they remain valid when the graph lies below the horizontal axis.