Free math tool
Riemann Integral Calculator
Calculate and visualize definite integrals using left, right, and midpoint Riemann sums with step-by-step explanations.
Visual Integral Calculator
Riemann Integral
Enter a function, choose an interval, and approximate the definite integral using Riemann sums.
Integral
\[
\int_{0}^{2} x^2\,dx
\]
Try:
Increase the number of rectangles to see how the Riemann sum approaches the definite integral.
∫
Your integral will appear here
Enter a function and interval to visualize its Riemann sum and calculate the approximate area.
Result
Riemann Sum Approximation
Integral
Step 1 — Divide the interval
The interval is divided into — equal subintervals.
Step 2 — Choose sample points
Step 3 — Build the Riemann sum
Step 4 — Approximate the area
Visual explanation
See the Riemann Sum
Each rectangle approximates a small piece of the signed area between the function and the x-axis.
How it works
From rectangles to an integral
Divide the interval
Split the interval [a, b] into n smaller pieces of equal width.
Build rectangles
Use the function value at the left, right, or midpoint of each subinterval as the rectangle height.
Add their areas
The signed rectangle areas are added to form the Riemann sum approximation.
Let n grow
As the rectangles become thinner, the Riemann sum approaches the definite integral when the function is integrable.
Definition
Riemann Integral Formula
\[
\int_a^b f(x)\,dx
=
\lim_{n\to\infty}
\sum_{i=1}^{n}
f(x_i^*)\Delta x
\]
where \(\Delta x = \frac{b-a}{n}\) and \(x_i^*\) is the sample point selected inside each subinterval.
Simple workflow
How to use this tool
- Enter or select a mathematical problem.
- Calculate and review every step.
- Copy, print, or download your work.
Built for understanding
Why the steps matter
The result is only the finish line. Each transformation shows the method so you can check the reasoning and solve similar problems yourself.
Questions
Frequently asked questions
Is this tool free?
Yes. All five launch tools are free and require no account.
Can I use the result in a worksheet?
Yes. Use print or PDF for a clean page, and use SVG or high-resolution PNG where graph export is available.
What happens when an input is unsupported?
The tool gives a clear message. It never invents a solution.