Table of Contents
Riemann sums are a systematic way to approximate a definite integral by adding up areas of simple shapes—usually rectangles—under (or above) the graph of a function. In this chapter, we focus on how Riemann sums are built, the main types of Riemann sums, and how they connect conceptually and symbolically to definite integrals.
We will always assume that $f$ is a real-valued function defined on a closed interval $[a,b]$.
Partitions of an interval
To form a Riemann sum over $[a,b]$, we first break the interval into smaller subintervals.
A partition $P$ of $[a,b]$ is a finite set of points
$$
P:\quad a = x_0 < x_1 < x_2 < \dots < x_{n-1} < x_n = b.
$$
These points divide $[a,b]$ into $n$ subintervals:
$$
[x_0,x_1],\ [x_1,x_2],\ \dots,\ [x_{n-1},x_n].
$$
The length (or width) of the $i$th subinterval is
$$
\Delta x_i = x_i - x_{i-1}, \quad i = 1,2,\dots,n.
$$
The norm (or mesh size) of the partition $P$ is
$$
\|P\| = \max_{1 \le i \le n} \Delta x_i.
$$
Conceptually, $\|P\|$ measures the size of the largest subinterval; as we refine the partition (use more and smaller subintervals), $\|P\|$ gets smaller.
A common special case is a uniform partition, where all subintervals have equal length:
$$
\Delta x_1 = \Delta x_2 = \dots = \Delta x_n = \Delta x = \frac{b-a}{n},
$$
and the partition points are
$$
x_i = a + i\Delta x, \quad i = 0,1,\dots,n.
$$
Uniform partitions make formulas and computations much simpler, and are standard in examples.
Sample points
Once we have a partition, we choose a point inside each subinterval to sample the function. These are called sample points.
For each subinterval $[x_{i-1},x_i]$, choose a point
$$
c_i \in [x_{i-1}, x_i], \quad i = 1,2,\dots,n.
$$
The value $f(c_i)$ will be used as the “height” of a rectangle over the $i$th subinterval. Different rules for picking $c_i$ lead to different kinds of Riemann sums (left, right, midpoint, etc.).
In symbols, a tagged partition is a partition together with a choice of sample point in each subinterval:
$$
\{( [x_{i-1},x_i], c_i )\}_{i=1}^n.
$$
General Riemann sum
Given
- a function $f$ defined on $[a,b]$,
- a partition $P: a = x_0 < x_1 < \dots < x_n = b$,
- sample points $c_i \in [x_{i-1},x_i]$,
the corresponding Riemann sum of $f$ on $[a,b]$ is
$$
S(f,P,\{c_i\}) = \sum_{i=1}^n f(c_i)\,\Delta x_i
= \sum_{i=1}^n f(c_i)\,(x_i - x_{i-1}).
$$
Geometrically, this sum adds the areas of rectangles:
- base of the $i$th rectangle: length $\Delta x_i$,
- height of the $i$th rectangle: $f(c_i)$.
If $f(c_i) \ge 0$, the rectangle lies above the $x$-axis; if $f(c_i) < 0$, the rectangle lies below the $x$-axis, contributing a negative amount. Thus, Riemann sums approximate the signed area under the graph of $y=f(x)$ over $[a,b]$.
In the frequently used uniform case with $\Delta x = \dfrac{b-a}{n}$ and $x_i = a + i\Delta x$, the general Riemann sum becomes
$$
S = \sum_{i=1}^n f(c_i)\,\Delta x,\quad
\text{with } c_i \in [x_{i-1},x_i].
$$
Types of Riemann sums (choice of sample points)
The general definition allows any choice of sample points in each subinterval. In practice, several standard choices are especially common.
Throughout this section, assume a uniform partition, so that
$$
\Delta x = \frac{b-a}{n},\quad x_i = a + i\Delta x.
$$
Left Riemann sum
For the left Riemann sum, the sample point in each subinterval is the left endpoint:
$$
c_i = x_{i-1},\quad i=1,\dots,n.
$$
The left Riemann sum is
$$
L_n = \sum_{i=1}^n f(x_{i-1})\,\Delta x.
$$
Equivalently, using $x_{i-1} = a + (i-1)\Delta x$,
$$
L_n = \sum_{i=1}^n f\big(a + (i-1)\Delta x\big)\,\Delta x.
$$
Interpretation: for each subinterval, we construct a rectangle whose height is the function value at the left end of the interval.
Right Riemann sum
For the right Riemann sum, the sample point in each subinterval is the right endpoint:
$$
c_i = x_i,\quad i=1,\dots,n.
$$
The right Riemann sum is
$$
R_n = \sum_{i=1}^n f(x_i)\,\Delta x
= \sum_{i=1}^n f\big(a + i\Delta x\big)\,\Delta x.
$$
Here, each rectangle takes its height from the right end of the subinterval.
Midpoint Riemann sum
For the midpoint Riemann sum, the sample point is the midpoint of each subinterval:
$$
c_i = \frac{x_{i-1} + x_i}{2}.
$$
In the uniform case,
$$
c_i = a + \left(i - \frac{1}{2}\right)\Delta x.
$$
The midpoint Riemann sum is
$$
M_n = \sum_{i=1}^n f\left(\frac{x_{i-1} + x_i}{2}\right)\Delta x
= \sum_{i=1}^n f\!\left(a + \left(i-\frac{1}{2}\right)\Delta x\right)\Delta x.
$$
Midpoint sums often give better approximations than left or right sums for the same number of subintervals, especially when $f$ is reasonably smooth.
Other choices: upper and lower sums
For a bounded function $f$ on $[a,b]$, we can also define special Riemann sums based on maximum and minimum values on each subinterval.
On each $[x_{i-1},x_i]$:
- let $M_i = \sup\{f(x) : x \in [x_{i-1},x_i]\}$ (the supremum/maximum),
- let $m_i = \inf\{f(x) : x \in [x_{i-1},x_i]\}$ (the infimum/minimum).
The upper sum $U(f,P)$ and lower sum $L(f,P)$ with respect to the partition $P$ are
$$
U(f,P) = \sum_{i=1}^n M_i\,\Delta x_i,\qquad
L(f,P) = \sum_{i=1}^n m_i\,\Delta x_i.
$$
These are special Riemann sums where the heights are, respectively, the largest and smallest values of $f$ on each subinterval. They play a key role in a more rigorous (Dar\-boux) approach to integration, but the basic computational idea remains: approximate area by rectangles.
Notation and connection to the definite integral
The definite integral $\displaystyle \int_a^b f(x)\,dx$ is defined (in the Riemann sense) as the limit of Riemann sums as the partition gets finer and finer, provided this limit exists and is independent of how we choose the sample points.
Symbolically, for a uniform partition with $\Delta x = \dfrac{b-a}{n}$ and some choice of $c_i \in [x_{i-1},x_i]$, we often write
$$
\int_a^b f(x)\,dx
= \lim_{n \to \infty} \sum_{i=1}^n f(c_i)\,\Delta x,
$$
when this limit exists and is the same for all valid choices of $c_i$.
In more general (non-uniform) terms, one can express the idea as:
$$
\int_a^b f(x)\,dx
= \lim_{\|P\| \to 0} \sum_{i=1}^n f(c_i)\,\Delta x_i,
$$
where the limit is taken as the norm $\|P\|$ of the partition tends to $0$ and the choice of $c_i \in [x_{i-1},x_i]$ is arbitrary.
The key conceptual link:
- A Riemann sum is a finite sum that approximates the definite integral.
- The definite integral is the limit of these sums as the largest subinterval width shrinks to zero.
The integral sign $\int$ together with $dx$ compactly encodes this limiting process over Riemann sums.
Expressing integrals as Riemann sum limits
Being able to recognize an expression as a Riemann sum, and rewrite it as a definite integral (or vice versa), is a fundamental skill.
A typical Riemann-sum expression for $f$ on $[a,b]$ has the form
$$
\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\,\Delta x,
$$
where
- $\Delta x = \dfrac{b-a}{n}$,
- $x_i^$ is some sample point in $[x_{i-1},x_i]$ (e.g., $x_i^ = a + i\Delta x$ or $x_i^ = a + (i-1)\Delta x$ or $x_i^ = a + (i - \tfrac{1}{2})\Delta x$).
In such a case, when the limit exists, we interpret
$$
\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\,\Delta x
= \int_a^b f(x)\,dx.
$$
Conversely, starting from $\displaystyle \int_a^b f(x)\,dx$, a standard Riemann-sum approximation with $n$ subintervals is
$$
\sum_{i=1}^n f\big(a + i\Delta x\big)\,\Delta x
\quad\text{or}\quad
\sum_{i=1}^n f\big(a + (i-1)\Delta x\big)\,\Delta x
\quad\text{or}\quad
\sum_{i=1}^n f\!\left(a + \left(i-\frac{1}{2}\right)\Delta x\right)\Delta x,
$$
depending on whether we use right, left, or midpoint sums.
The structure to look for is:
- A limit as $n \to \infty$,
- A sum $\sum_{i=1}^n$,
- A factor that plays the role of $\Delta x$,
- A function evaluated at points depending on $i$ and $n$, representing $x_i^*$.
Once you identify $a$, $b$, and $f$, you can rewrite the expression as $\displaystyle \int_a^b f(x)\,dx$.
Approximation and error (informal view)
In practice, we often use Riemann sums with a finite $n$ to approximate a definite integral numerically. Key ideas, without detailed error formulas:
- More subintervals (larger $n$) generally yield a better approximation, because the rectangles follow the curve more closely as $\Delta x$ becomes smaller.
- If $f$ is continuous and well-behaved, Riemann sums converge to the exact integral as $n \to \infty$.
- The midpoint sum often gives a more accurate approximation than the left or right sum for the same $n$, especially when $f$ is smooth.
One common strategy is to approximate
$$
\int_a^b f(x)\,dx \approx S_n
$$
for some chosen Riemann sum $S_n$, and increase $n$ until successive approximations change very little.
A more precise analysis of error requires additional tools and is treated elsewhere; here the focus is on the Riemann sum construction itself.
Summary of key formulas
Let $f$ be defined on $[a,b]$, and let $n$ be a positive integer. For a uniform partition:
- Partition points:
$$
\Delta x = \frac{b-a}{n},\quad x_i = a + i\Delta x,\quad i=0,1,\dots,n.
$$ - General Riemann sum:
$$
S_n = \sum_{i=1}^n f(c_i)\,\Delta x,\quad c_i \in [x_{i-1},x_i].
$$ - Left Riemann sum:
$$
L_n = \sum_{i=1}^n f(x_{i-1})\,\Delta x
= \sum_{i=1}^n f\big(a + (i-1)\Delta x\big)\,\Delta x.
$$ - Right Riemann sum:
$$
R_n = \sum_{i=1}^n f(x_i)\,\Delta x
= \sum_{i=1}^n f\big(a + i\Delta x\big)\,\Delta x.
$$ - Midpoint Riemann sum:
$$
M_n = \sum_{i=1}^n f\left(\frac{x_{i-1} + x_i}{2}\right)\Delta x
= \sum_{i=1}^n f\!\left(a + \left(i-\frac{1}{2}\right)\Delta x\right)\Delta x.
$$ - Definite integral as limit of Riemann sums (uniform case):
$$
\int_a^b f(x)\,dx
= \lim_{n \to \infty} \sum_{i=1}^n f(c_i)\,\Delta x,
$$
with $c_i \in [x_{i-1},x_i]$ any sample points, when the limit exists and is independent of their choice.
These formulas encapsulate the Riemann-sum approach to definite integrals: approximate a complicated area by many simple rectangles, then let the number of rectangles grow without bound.