How to Calculate a Riemann Sum: Step-by-Step Guide with Examples

Quick takeaway: Need to calculate a Riemann sum fast? Jump straight to our free Riemann Sum Calculator for instant results, or keep reading to master the method by hand.

Why Learning to Calculate Riemann Sums Actually Matters #

If you're taking calculus, you've probably heard the term "Riemann sum" thrown around like it's some ancient math ritual. But here's the truth: Riemann sums are how we approximate the area under a curve—a practical skill used everywhere from calculating distance traveled to estimating total costs over time.

Whether you're solving homework problems, preparing for an exam, or just trying to understand what integration really means, knowing how to calculate a Riemann sum gives you the foundation for understanding calculus at a deeper level.

(Need help with other types of sums? Check out our Sum of Series Calculator and Infinite Sum Calculator.)

Table of Contents #

What Is a Riemann Sum?
Types of Riemann Sums
How to Calculate a Riemann Sum (Step-by-Step)
The Riemann Sum Formula
Real-World Example: Calculating Distance
Common Mistakes to Avoid
Using a Riemann Sum Calculator
Frequently Asked Questions (FAQ)

What Is a Riemann Sum? #

A Riemann sum is a method for approximating the total value (or area) under a curve by dividing it into rectangles and adding up their areas. It's named after the German mathematician Bernhard Riemann, who formalized the concept in the 1800s.

Think of it this way: If you want to find the area under a wavy curve but don't have the exact formula, you can chop the area into thin vertical slices (rectangles), calculate the area of each rectangle, then add them all together. The more rectangles you use, the better your approximation.

Key components:

A function $f(x)$ that defines your curve
An interval $[a, b]$ over which you're calculating
A number of subintervals $n$ that determines how finely you divide the area
Sample points (left, right, or midpoint) that determine the height of each rectangle

Riemann sums are the foundation of definite integrals in calculus. When you let the number of rectangles approach infinity, the Riemann sum becomes the exact integral.

Types of Riemann Sums #

There are three main types of Riemann sums, each using a different method to determine rectangle height:

Left Riemann Sum #

The height of each rectangle is determined by the function value at the left endpoint of each subinterval. This method tends to underestimate the area if the function is increasing, and overestimate if it's decreasing.

Right Riemann Sum #

The height is determined by the function value at the right endpoint of each subinterval. This is the mirror opposite of the left sum—it overestimates for increasing functions and underestimates for decreasing ones.

Midpoint Riemann Sum #

The height is determined by the function value at the midpoint of each subinterval. This method generally gives a better approximation than left or right sums because it balances out the over- and under-estimation.

Trapezoidal Rule (Bonus Method) #

While not technically a Riemann sum, the trapezoidal rule averages the left and right endpoints to form trapezoids instead of rectangles. It often provides even better accuracy.

Method	Sample Point	Best For	Typical Accuracy
Left Riemann Sum	Left endpoint	Quick estimates	Moderate
Right Riemann Sum	Right endpoint	Quick estimates	Moderate
Midpoint Riemann Sum	Midpoint	Better accuracy	Good
Trapezoidal Rule	Average of endpoints	Highest accuracy	Excellent

How to Calculate a Riemann Sum (Step-by-Step) #

Let's walk through a complete example using a left Riemann sum.

Problem: Approximate the area under the curve $f(x) = x^2$ from $x = 0$ to $x = 4$ using 4 rectangles.

Step 1: Define the Function and Interval #

Function: $f(x) = x^2$
Interval: $[a, b] = [0, 4]$
Number of rectangles: $n = 4$

Step 2: Calculate the Width of Each Rectangle #

The width of each rectangle, denoted $\Delta x$ , is:

\Delta x = \frac{b - a}{n} = \frac{4 - 0}{4} = 1

Each rectangle has a width of 1 unit.

Step 3: Identify the Sample Points #

For a left Riemann sum, use the left endpoint of each subinterval:

$x_0 = 0$
$x_1 = 1$
$x_2 = 2$
$x_3 = 3$

Step 4: Calculate the Height of Each Rectangle #

Plug each sample point into the function:

$f(x_0) = f(0) = 0^2 = 0$
$f(x_1) = f(1) = 1^2 = 1$
$f(x_2) = f(2) = 2^2 = 4$
$f(x_3) = f(3) = 3^2 = 9$

Step 5: Sum It All Up #

The left Riemann sum is:

L_4 = \sum_{i=0}^{3} f(x_i) \cdot \Delta x = [0 + 1 + 4 + 9] \times 1 = 14

Result: The approximate area is 14 square units.

For comparison, the exact area (using integration) is:

\int_0^4 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^4 = \frac{64}{3} \approx 21.33

Our left Riemann sum underestimated by about 7.33 units because the function is increasing. Using more rectangles would improve accuracy.

The Riemann Sum Formula #

The general formula for a Riemann sum is:

R_n = \sum_{i=1}^{n} f(x_i^*) \cdot \Delta x

Where:

$n$ = number of subintervals (rectangles)
$\Delta x = \frac{b-a}{n}$ = width of each rectangle
$x_i^*$ = the sample point in the $i$ -th subinterval (left, right, or midpoint)
$f(x_i^*)$ = height of the $i$ -th rectangle

For specific methods:

Left sum: $x_i^* = a + (i-1)\Delta x$
Right sum: $x_i^* = a + i\Delta x$
Midpoint sum: $x_i^* = a + (i - 0.5)\Delta x$

Real-World Example: Calculating Distance #

Suppose you're driving, and your speed varies over time according to $v(t) = 2t + 3$ (in meters per second), where $t$ is time in seconds. How far do you travel between $t = 0$ and $t = 6$ seconds?

We can approximate this using a right Riemann sum with $n = 3$ rectangles.

Step 1: Calculate $\Delta t$

\Delta t = \frac{6 - 0}{3} = 2 \text{ seconds}

Step 2: Identify right endpoints

$t_1 = 2$ , $t_2 = 4$ , $t_3 = 6$

Step 3: Calculate velocity at each endpoint

$v(2) = 2(2) + 3 = 7$ m/s
$v(4) = 2(4) + 3 = 11$ m/s
$v(6) = 2(6) + 3 = 15$ m/s

Step 4: Calculate the Riemann sum

R_3 = [7 + 11 + 15] \times 2 = 33 \times 2 = 66 \text{ meters}

Result: You traveled approximately 66 meters in 6 seconds.

(For complex calculations, use our Riemann Sum Calculator to verify your work.)

Common Mistakes to Avoid #

Mistake	Why It Happens	How to Fix It
Forgetting to multiply by $\Delta x$	You calculate the sum of heights but forget the width.	Always include $\Delta x$ in your final calculation.
Using the wrong sample point	Mixing up left, right, and midpoint formulas.	Double-check which method you're using and stick to it.
Counting rectangles incorrectly	Off-by-one errors when determining $n$ .	Clearly list all subintervals before calculating.
Rounding too early	Premature rounding compounds errors.	Keep full precision until the final step.
Not checking your answer	Small mistakes multiply across many rectangles.	Verify with a calculator or by trying a different method.

Using a Riemann Sum Calculator #

When you're short on time or dealing with complex functions, an online calculator is your best friend. Our Riemann Sum Calculator lets you:

Choose between left, right, and midpoint methods
Input any function and interval
Adjust the number of rectangles for better accuracy
Get instant, step-by-step results

It's perfect for checking homework, studying for exams, or just avoiding tedious hand calculations. And it's 100% free.

For related calculations, explore our Summation Calculator and Average Calculator.

Frequently Asked Questions (FAQ) #

Q1: What is the difference between left and right Riemann sums?
The left Riemann sum uses the function value at the left endpoint of each subinterval to determine rectangle height, while the right sum uses the right endpoint. For increasing functions, the left sum underestimates and the right sum overestimates (and vice versa for decreasing functions).

Q2: How many rectangles should I use for an accurate approximation?
More rectangles always give better accuracy. For homework, $n = 4$ to $n = 10$ is typical. For practical use, $n = 100$ or more provides near-exact results. Online calculators handle large $n$ values effortlessly.

Q3: When should I use the midpoint rule instead of left or right?
Use the midpoint rule when you want better accuracy without doing significantly more work. It generally has lower error than left or right sums for the same number of rectangles.

Q4: Is the Riemann sum the same as the integral?
Not exactly. The Riemann sum is an approximation. As the number of rectangles approaches infinity ( $n \to \infty$ ), the Riemann sum approaches the exact value of the definite integral.

Q5: Can I use Riemann sums for functions that aren't continuous?
Yes, but with caution. Riemann sums work best on continuous functions. For functions with jumps or discontinuities, you may need to adjust your method or use more advanced techniques.

Final Takeaway #

Learning how to calculate a Riemann sum is one of those skills that seems tricky at first but becomes second nature with practice. Start by mastering the basic steps—define your interval, calculate $\Delta x$ , pick your sample points, and sum it all up. Then move on to more complex functions and larger values of $n$ .

For everyday calculations, bookmark our Riemann Sum Calculator and other tools at sumcalculator.org.

And if you want to dive deeper into series and summation techniques, check out our guide on How to Sum a Series. We're here to make math less painful—one rectangle at a time.