Computing Riemann Sums

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From Rectangles to Areas

How do you find the area of a region with curved boundaries? You can’t use a simple formula like length times width. The brilliant idea behind calculus is to approximate the curved region with rectangles, then make the approximation better and better.

A Riemann sum is exactly this: the total area of rectangles that approximate the region under a curve. By increasing the number of rectangles, the approximation improves. In the limit, as we use infinitely many infinitely thin rectangles, we get the exact area.

The key insight: A Riemann sum isn’t just a numerical approximation. It’s a fundamental way of thinking about accumulation. Every integral you’ll ever compute is, at its heart, a limit of Riemann sums.

Prerequisite Map

Quick Reference

Key Concepts

Setting Up a Riemann Sum

To approximate $\int_a^b f(x)\,dx$ using $n$ rectangles:

Step 1: Divide the interval $[a,b]$ into $n$ equal subintervals.

The width of each subinterval is: $$\Delta x = \frac{b - a}{n}$$

The endpoints are: $$x_0 = a, \quad x_1 = a + \Delta x, \quad x_2 = a + 2\Delta x, \quad \ldots, \quad x_n = b$$

In general: $x_i = a + i \cdot \Delta x$

Step 2: Choose sample points in each subinterval $[x_{i-1}, x_i]$.

Step 3: Form the sum of rectangle areas: $$\sum_{i=1}^{n} f(x_i^*) \cdot \Delta x$$

where $x_i^*$ is the sample point in the $i$th subinterval.

Types of Riemann Sums

Visualizing Riemann Sums

For $f(x) = x^2$ on $[0, 2]$ with $n = 4$ (right endpoints):

    y
    |
  4 +                    ■
    |                ■■■■■
  3 +                ■
    |            ■■■■■
  2 +            ■
    |        ■■■■■
  1 +    ■■■■■
    |■■■■■
  0 +----+----+----+----+→ x
    0   0.5   1   1.5   2

Each rectangle has width $\Delta x = 0.5$ and height $f(x_i)$ where $x_i$ is the right endpoint.

The Riemann Sum Formula

$$R_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} f\left(a + i \cdot \frac{b-a}{n}\right) \cdot \frac{b-a}{n}$$

Important observation: For an increasing function:

• $L_n$ underestimates the area (rectangles fit inside)
• $R_n$ overestimates the area (rectangles extend beyond)

For a decreasing function, the opposite is true.

Physical Interpretation

Riemann sums model accumulated quantities:

• Distance from velocity: If $v(t)$ is velocity, then $\sum v(t_i^*) \Delta t$ approximates total distance
• Mass from density: If $\rho(x)$ is linear density, then $\sum \rho(x_i^*) \Delta x$ approximates total mass
• Work from force: If $F(x)$ is force, then $\sum F(x_i^*) \Delta x$ approximates total work

Practice Problems

Common Misconceptions

Mastery Checklist

• I can compute $\Delta x$ given $[a, b]$ and $n$
• I can list the endpoints $x_0, x_1, \ldots, x_n$ of the subintervals
• I can identify left, right, and midpoint sample points
• I can compute left, right, and midpoint Riemann sums for specific functions
• I can determine whether a Riemann sum overestimates or underestimates based on whether $f$ is increasing or decreasing
• I can write a Riemann sum in sigma notation

Mental Model

The Staircase Approximation: Imagine building a staircase that follows a curved hill. Each step (rectangle) has a fixed width but varies in height based on the hill at some point in that interval. A few wide steps give a rough approximation; many narrow steps follow the curve more closely. The Riemann sum is the total vertical material in your staircase. As steps get narrower and more numerous, your staircase becomes indistinguishable from the smooth hill.

Connections

Looking back:

• Sigma Notation provides the language for writing Riemann sums compactly
• Function Evaluation is needed to compute $f(x_i^*)$ at sample points

Looking ahead:

• The Definite Integral Definition is the limit of Riemann sums as $n \to \infty$
• The Fundamental Theorem of Calculus gives a faster way to evaluate integrals
• Numerical methods like the Trapezoidal Rule and Simpson’s Rule are refinements of Riemann sums

Last updated: 2026-01-22