Riemann Sum Calculator (Left, Right, Midpoint, Trapezoid)

Use this free Riemann sum calculator to approximate a definite integral. Choose left endpoints, right endpoints, midpoints, or the trapezoidal rule, set the number of subintervals, and see the value, a graph of the rectangles, and the steps.

How it works

The interval [a, b] is split into n equal pieces of width Δx = (b − a)/n. Each piece contributes its width times a sample height; the method decides which sample point is used.

How to use it

1. Enter f(x), the interval a and b, and n.
2. Pick the method.
3. Press Compute sum to see the value, the rectangles, and the estimated error.

Related: the Integral Calculator and more free math calculators.

Frequently asked questions

How do you calculate a Riemann sum?

Split [a, b] into n equal pieces of width Δx = (b − a)/n, pick a sample point in each piece, and add up f(sample) × Δx. For f(x) = x² on [0, 2] with n = 4 and right endpoints, Δx = 0.5 and the sum is 0.5 × (0.25 + 1 + 2.25 + 4) = 3.75. The exact integral is 8/3 ≈ 2.67, so this sum overestimates.

How do you do a left Riemann sum?

Use the left endpoint of each subinterval as the rectangle’s height. With f(x) = x² on [0, 2] and n = 4, the left endpoints are 0, 0.5, 1 and 1.5, so the sum is 0.5 × (0 + 0.25 + 1 + 2.25) = 1.75. For an increasing function like this one, a left sum always underestimates the area.

How do you find a midpoint Riemann sum?

Use the middle of each subinterval as the sample point. For f(x) = x² on [0, 2] with n = 4, the midpoints are 0.25, 0.75, 1.25 and 1.75, so the sum is 0.5 × (0.0625 + 0.5625 + 1.5625 + 3.0625) = 2.625. That is much closer to the exact value 8/3 ≈ 2.667 than either the left or right sum.

How do you calculate a Riemann sum with a table?

When only a table is given, use the listed x values as endpoints and multiply each height by the width of its subinterval, which may not be equal. For x = 0, 2, 5, 6 with f(x) = 3, 4, 7, 8, a left sum is 3(2) + 4(3) + 7(1) = 25, and a right sum is 4(2) + 7(3) + 8(1) = 37.

How do you do a Riemann sum on a TI-84 calculator?

The TI-84 has no Riemann sum button, but sum( and seq( from the LIST menu do the job. For a right sum of x² on [0, 2] with n = 4, enter sum(seq((0.5K)²×0.5, K, 1, 4)), which returns 3.75. Run K from 0 to 3 for a left sum, or replace 0.5K with 0.5K − 0.25 for midpoints.

Is Riemann sum Calc 1 or Calc 2?

Riemann sums are usually introduced in Calculus 1, when the definite integral is first defined as a limit of sums. They return in Calculus 2 alongside numerical methods like the trapezoidal rule and in applications such as area and volume. In AP Calculus, Riemann sums appear in both the AB and BC courses.

How to use the Riemann Sum Calculator for homework

The Riemann Sum Calculator is most useful when you treat it as a learning check, not just a shortcut to the final answer. Start by copying the original problem carefully, including signs, exponents, decimal points, fractions, parentheses, and units. Then enter the values in the same order the problem gives them. A small typing change can completely change the result, especially in algebra, statistics, geometry, and probability problems.

Before you press the button to calculate, make a quick estimate or prediction. The estimate does not need to be exact. Its job is to help you notice impossible answers. If a distance becomes negative, a probability is bigger than 1, an angle looks too large, or a decimal point seems misplaced, go back and check the input before trusting the final result.

Before you enter the problem

• Rewrite the problem in a clean line so every value is easy to see.
• Use parentheses around grouped expressions, especially in fractions and exponents.
• Keep units with the numbers while you work, even if the calculator only asks for the numbers.
• Check whether the problem wants an exact value, a decimal approximation, or a rounded answer.
• Look for restrictions such as positive values only, a chosen interval, or a required domain.

How to read the result

After the calculator gives a result, read more than the final line. If steps, tables, graphs, or intermediate values are shown, use them to understand how the answer was built. That is especially important when you are studying for a quiz or test, because teachers often give more credit for a correct process than for an unsupported number.

Try to identify the main idea behind the result. For example, ask yourself which formula was used, which operation changed the expression, which value controlled the graph, or which assumption made the answer possible. When you can explain that idea in your own words, the tool has helped you learn the skill instead of only checking one problem.

Common mistakes to avoid

Turn the answer into practice

One good way to study is to solve the problem by hand first, then use the Riemann Sum Calculator to check your work. If your answer is wrong, do not erase everything immediately. Find the first line where your work stops matching the calculator’s logic. That line is usually where the real misunderstanding happened.

1. Work the problem on paper and circle your final answer.
2. Use the calculator to check the result.
3. If the answers match, write one sentence explaining the method.
4. If they do not match, compare each step until you find the first difference.
5. Redo a similar problem without the tool to make sure the correction sticks.

When to use a calculator and when to work by hand

Use the Riemann Sum Calculator when you want to check a long calculation, explore a pattern, test a graph, or confirm a result after practicing. Work by hand when the assignment asks for steps, when you are learning a new method, or when the test will not allow a digital tool. The strongest students use both: hand work to build understanding and calculators to check accuracy.

If you are preparing for a timed test, practice some problems without the tool and some with it. That balance helps you build speed while still understanding the math. Over time, the goal is to need the calculator less often for routine steps and use it more strategically for checking, exploring, and confirming your reasoning.

Quick accuracy checklist

Before you leave the page, use the Riemann Sum Calculator result to answer three quick questions: What was the input? What rule, formula, or method connects the input to the result? What would change if one number in the problem changed? These questions turn a single calculator answer into a reusable math skill.

For homework, write the original problem, the calculator’s result, and one short explanation of the method in your notebook. For test review, cover the result and try to reproduce the setup on your own. If you can set up a similar problem without looking back at the calculator, you are much closer to owning the skill.