Partitions and Upper/Lower Sums 📘

Introduction

students, this lesson starts the core of Riemann integration, one of the main ways mathematicians measure the total effect of a function over an interval. The big idea is simple: instead of trying to find area all at once, we cut an interval into small pieces and estimate the function using rectangles. As the pieces get smaller, the estimate becomes more accurate. 😊

Learning objectives

By the end of this lesson, students, you should be able to:

explain what a partition of an interval is,
describe upper sums and lower sums,
use these ideas to estimate the area under a graph,
understand why these tools matter for deciding whether a function is Riemann integrable,
connect this lesson to the larger topic of Riemann Integration I.

A helpful real-world picture is this: imagine you want to estimate how much water is in a lake with an uneven bottom. You cannot measure every point exactly, so you divide the lake into sections and use simple measurements. Partitions and upper/lower sums do something similar for functions over an interval.

Partitions: dividing an interval into pieces

A partition of an interval $[a,b]$ is a finite set of points

$$P=\{x_0,x_1,x_2,\dots,x_n\}$$

with

$$a=x_0<x_1<x_2<\cdots<x_n=b.$$

These points split the interval $[a,b]$ into smaller subintervals:

$$[x_0,x_1], [x_1,x_2], \dots, [x_{n-1},x_n].$$

Each subinterval has length

$$\Delta x_i=x_i-x_{i-1}.$$

The partition helps us study a function $f$ on $[a,b]$ piece by piece.

Why partitions matter

If $f$ changes a lot across a big interval, one rectangle may be a poor estimate. But if the interval is cut into many small parts, the function is easier to approximate on each part. The more finely we divide the interval, the better our estimates usually become.

A partition is called refined when more points are added, making the subintervals smaller. If $Q$ contains all the points of $P$ and possibly more, then $Q$ is a refinement of $P$.

Example

Suppose $[a,b]=[0,4]$ and we choose

$$P=\{0,1,3,4\}.$$

Then the subintervals are $[0,1]$, $[1,3]$, and $[3,4]$. Their lengths are $1$, $2$, and $1$.

This partition is not evenly spaced, and that is completely fine. Partitions do not have to be equal in length.

Upper sums and lower sums

Now let $f$ be a bounded function on $[a,b]$. For each subinterval $[x_{i-1},x_i]$, we look at the largest and smallest values of $f$ on that piece.

Define

$$M_i=\sup\{f(x):x\in[x_{i-1},x_i]\}$$

and

$$m_i=\inf\{f(x):x\in[x_{i-1},x_i]\}.$$

Here, $\sup$ means supremum or least upper bound, and $\inf$ means infimum or greatest lower bound.

The upper sum of $f$ with respect to partition $P$ is

$$U(f,P)=\sum_{i=1}^n M_i\Delta x_i,$$

and the lower sum is

$$L(f,P)=\sum_{i=1}^n m_i\Delta x_i.$$

What these sums mean

The upper sum uses the highest possible value of $f$ on each subinterval, so it gives an estimate that is at least as large as the actual area idea.
The lower sum uses the lowest possible value of $f$ on each subinterval, so it gives an estimate that is at most as large as the actual area idea.

If $f(x)\ge 0$, then upper and lower sums can be thought of as “too high” and “too low” rectangle approximations of area. Even when $f$ takes negative values, the same definitions still work; the geometry becomes more subtle, but the algebra is reliable.

A simple example with a nonnegative function

Let $f(x)=x$ on $[0,2]$ and use the partition

$$P=\{0,1,2\}.$$

On $[0,1]$, the function ranges from $0$ to $1$, so

$$m_1=0,\quad M_1=1.$$

On $[1,2]$, it ranges from $1$ to $2$, so

$$m_2=1,\quad M_2=2.$$

Since each subinterval has length $1$,

$$L(f,P)=0\cdot1+1\cdot1=1,$$

and

$$U(f,P)=1\cdot1+2\cdot1=3.$$

The exact area under $y=x$ from $0$ to $2$ is

$$\int_0^2 x\,dx=2,$$

so the lower sum is below the true value and the upper sum is above it. This is exactly what we want! The true value is trapped between them:

$$L(f,P)\le 2\le U(f,P).$$

A real-world interpretation

Imagine estimating the cost of electricity usage over time when the rate changes throughout the day. If you use the lowest rate in each time block, you get a lower estimate. If you use the highest rate, you get an upper estimate. A finer partition of the day gives a better estimate. ⚡

Properties of upper and lower sums

Upper and lower sums have important patterns.

1. Lower sums are always less than or equal to upper sums

For any partition $P$,

$$L(f,P)\le U(f,P).$$

This is because on each subinterval,

$$m_i\le M_i,$$

and multiplying by the positive length $\Delta x_i$ preserves the inequality.

2. Refining a partition improves control

If $Q$ is a refinement of $P$, then

$$L(f,P)\le L(f,Q)\le U(f,Q)\le U(f,P).$$

This means that when we make the partition finer, lower sums usually go up and upper sums usually go down. They move closer together, which is a strong sign that the function may be integrable.

3. Bounded functions are necessary

To define $M_i$ and $m_i$ in a useful way, the function should be bounded on $[a,b]$. If a function is not bounded, the upper and lower sums may not behave well enough for Riemann integration.

How this connects to integrability

A bounded function $f$ on $[a,b]$ is Riemann integrable if the upper sums and lower sums can be made arbitrarily close.

A standard way to say this is: for every $\varepsilon>0$, there exists a partition $P$ such that

$$U(f,P)-L(f,P)<\varepsilon.$$

This means the gap between the overestimate and the underestimate can be made as small as we want.

Another equivalent idea is

$$\inf_P U(f,P)=\sup_P L(f,P).$$

When this happens, the common value is the Riemann integral of $f$ over $[a,b]$.

Why this is important

Not every bounded function is integrable, but many common ones are. For example:

continuous functions on closed intervals are integrable,
monotone functions on closed intervals are integrable,
step functions are integrable.

The partition and upper/lower sum approach gives a precise way to decide whether a function behaves nicely enough for integration.

Example of a function with a jump

Consider the function

$$f(x)=\begin{cases}0,&x<1,\\1,&x\ge 1,\end{cases}$$

on $[0,2]$.

If a partition includes the point $x=1$, then the interval pieces on each side are simple: the function is constant there, so upper and lower sums match on those pieces. If the partition does not include $1$, then one subinterval crosses the jump, and the upper and lower sums differ on that piece.

This function is still Riemann integrable because by choosing a partition that places a cut at the jump point and then refining the interval, the difference between upper and lower sums can be made small. The lesson here is that a function can have discontinuities and still be integrable, as long as the discontinuities are controlled in a suitable way.

Common mistakes to avoid

students, these ideas are easy to mix up, so watch for the following:

A partition is a set of points, not the rectangles themselves.
The upper sum uses suprema, not arbitrary high values.
The lower sum uses infima, not arbitrary low values.
Smaller subintervals do not automatically make the sums exact, but they often improve the estimate.
The upper and lower sums depend on both the function and the partition.

Conclusion

Partitions and upper/lower sums are the foundation of Riemann integration. By dividing an interval into smaller pieces, we can build controlled estimates of a function’s total behavior. Upper sums give overestimates, lower sums give underestimates, and integrability means these two estimates can be brought as close together as we want. This idea connects directly to the larger study of Riemann Integration I and prepares you for understanding the formal definition of the Riemann integral. 📚

Study Notes

A partition of $[a,b]$ is a finite ordered set of points $a=x_0<x_1<\cdots<x_n=b$.
The subinterval lengths are $\Delta x_i=x_i-x_{i-1}$.
For a bounded function $f$, define $M_i=\sup\{f(x):x\in[x_{i-1},x_i]\}$ and $m_i=\inf\{f(x):x\in[x_{i-1},x_i]\}$.
The upper sum is $U(f,P)=\sum_{i=1}^n M_i\Delta x_i$.
The lower sum is $L(f,P)=\sum_{i=1}^n m_i\Delta x_i$.
Always $L(f,P)\le U(f,P)$.
If $Q$ refines $P$, then $L(f,P)\le L(f,Q)\le U(f,Q)\le U(f,P)$.
A bounded function is Riemann integrable if for every $\varepsilon>0$ there is a partition $P$ such that $U(f,P)-L(f,P)<\varepsilon$.
Continuous functions on closed intervals are Riemann integrable.
Upper and lower sums are the bridge between graph-based intuition and the formal definition of the Riemann integral.