Integral & Area Under Curve
Drag the limits, choose the number of rectangles and compare Riemann sums with the exact area.
Drag the two orange handles on the x-axis to set the limits. Change the number of rectangles and compare with the exact area.
About the Integral & Area Under Curve
Free integral simulator. Drag the lower and upper limits, change the number of rectangles, choose left, right or midpoint sums, and compare with the exact integral. Built for math, the integral & area under curve runs instantly in your browser: change a setting or drag an object and the result updates at once, so you learn by trying things out rather than only reading about them.
Drag the limits, choose the number of rectangles and compare Riemann sums with the exact area. Use it to explore math ideas at your own pace, then check what you found against the key ideas further down this page.
How to use the Integral & Area Under Curve
- Use the controls to change Or type your own f(x), Rectangles n, Method. The simulation reacts instantly.
- Pick an option such as Left, Midpoint, Right, Trapezoid to switch modes or load an example.
- Where you see a glowing handle, object, weight or atom, drag it with your mouse or finger. Everything responds in real time.
- Watch the readouts and graphs update as you experiment, and compare what you see with the key ideas below.
Things to try
- Drag the upper limit and watch the area update.
- Increase the rectangles from 4 to 100 and see the error shrink.
- Compare left, right and midpoint sums for the same n.
- Choose sin(x) from 0 to π. What is the area?
Key ideas you can learn
- A definite integral is the signed area between the curve and the x-axis.
- Riemann sums approximate the area with rectangles. More rectangles gives a better answer.
- Midpoint sums are usually more accurate than left or right sums.
Where this is used in the real world
Integrals compute distance from speed, work from force, probabilities and areas, and are used across engineering and science.
Who is this simulation for?
Algebra, geometry, trigonometry and calculus students, teachers preparing demonstrations, and self-learners who want to see the maths move.
For teachers: project it on the board, let students predict what will happen, then run it together. For students: change one thing at a time and write down what changes.
Frequently asked questions
What is a Riemann sum?
An approximation of the area under a curve by adding up the areas of thin rectangles. As the rectangles get thinner the sum approaches the exact integral.
Why can an integral be negative?
Area below the x-axis counts as negative, so the integral is the area above minus the area below.
Is the Integral & Area Under Curve free to use?
Yes. It is completely free, with no signup, no download and no ads inside the simulation. It runs in your web browser.
Does the Integral & Area Under Curve work on a phone or tablet?
Yes. It uses touch as well as the mouse, so you can drag objects with your finger. A larger screen makes the controls easier to see.