Riemann Sum Calculator: Formula and Worked Example

A Riemann sum estimates the signed area under a function by adding the areas of rectangles across an interval. Enter the function’s endpoint values through its rectangle heights, along with the interval endpoints and number of rectangles, to calculate a left-endpoint Riemann sum and its rectangle width.

Formula

For an interval from a to b, divided into n equal subintervals, the left Riemann sum is

L_n = Δx [f(x₀) + f(x₁) + ··· + f(x_n−1)], where Δx = (b − a)/n and x_i = a + iΔx.

Here, a and b are the interval endpoints in x units, n is a positive whole-number rectangle count, and each f(x_i) is a function height in function units. The sum has units of function units multiplied by x units; for example, meters per second multiplied by seconds gives meters.

The calculator takes the heights directly as a list. Enter exactly n heights, in order, for the left endpoints. It calculates the rectangle width and multiplies that width by the sum of the heights.

Worked Example

Using the calculator’s defaults, the interval is [0, 4], the number of rectangles is 4, and the left-endpoint heights are 1, 2, 3, and 4.

1. Find the width: Δx = (4 − 0)/4 = 1.
2. Add the heights: 1 + 2 + 3 + 4 = 10.
3. Multiply the total height by the width: L₄ = 1 × 10 = 10.

The result is 10 function units × x units, and the rectangle width is 1 x unit. The displayed sum is rounded to two decimal places, so the result is 10.00.

What a Riemann Sum Measures

A Riemann sum is a finite approximation built from rectangles. Each rectangle has a width along the x-axis and a height taken from the function at a selected point in that subinterval. Adding the signed rectangle areas estimates the definite integral.

“Signed” matters: heights below the x-axis contribute negative area. A Riemann sum is therefore not necessarily the same as the total geometric area between the graph and the axis. To find total geometric area when a function crosses the axis, the intervals may need to be split at the crossings and the areas treated as positive.

For the defaults, if the heights come from f(x) = x + 1 on [0, 4], the left endpoints are 0, 1, 2, and 3, giving heights 1, 2, 3, and 4. The left sum is 10. The exact integral is 12, so this particular approximation is low.

Choosing Heights and Rectangle Rules

The rectangle rule determines where each rectangle samples the function. This calculator uses left endpoints. For equal-width intervals, the sample locations for a left sum are a, a + Δx, through b − Δx. Other common rules use different points:

• Right sum: samples at the right endpoint of each subinterval.
• Midpoint sum: samples at the midpoint of each subinterval.
• Trapezoidal rule: joins endpoint values with straight lines and sums trapezoid areas rather than rectangles.

These methods can produce different estimates for the same function and interval. For an increasing function, left-endpoint rectangles are typically below the curve and right-endpoint rectangles above it. For a decreasing function, that pattern reverses. If the function changes direction, there may be no single over-or-under pattern across the whole interval.

When to Use It and How to Improve an Estimate

Use a Riemann sum when an exact integral is difficult or unavailable, when function values are known only at sampled points, or when learning how integration is built from finite approximations. It also provides a simple numerical estimate when a function can be evaluated at chosen x-values.

With a fixed interval and a well-behaved function, increasing the number of rectangles usually improves the approximation because the widths shrink. It does not guarantee that every individual estimate moves closer in every case, especially for irregular or discontinuous functions. For a more dependable numerical answer, compare estimates using different rules or more rectangles, and check against an exact integral if one is available.

Common Errors and Limitations

• Wrong number of heights: A left sum with n rectangles needs exactly n left-endpoint heights, not n + 1 endpoint values.
• Mixing up the sample rule: Heights for a right or midpoint sum cannot be entered as though they were left-endpoint heights. The output would then represent a different sampling choice.
• Using inconsistent units: If x is measured in seconds and the function in meters per second, the result is in meters. Convert measurements first when the entries use incompatible units.
• Ignoring the sign: Negative function values must remain negative for a signed integral estimate; changing them to positive computes a different quantity.
• Rounding too early: Keep full-precision function values and rectangle widths during calculation, and round the final answer to the precision appropriate for the data.

The calculator assumes equal-width intervals and a list of already determined heights. It does not evaluate an equation, select a sampling rule other than left endpoints, or verify that supplied values correspond to the stated function and interval. If the interval is reversed, the signed sum is reversed as well.

Frequently Asked Questions

How do you calculate a Riemann sum?
Divide the interval width by the number of rectangles, add the selected function heights, then multiply the two results. The sample points used for the heights determine whether the sum is left, right, or midpoint.

How many function values do I enter?
For this left-sum calculator, enter one height per rectangle, so the list must contain n values. A separate calculation of the function’s values at all partition endpoints may contain n + 1 values, but the final endpoint is not used for a left sum.

Is a Riemann sum the same as an integral?
Not generally. A Riemann sum is a finite approximation. Under suitable conditions, sums with increasingly narrow subintervals converge to the definite integral.

Does a larger number of rectangles always give the exact answer?
No. More rectangles often improve an estimate for a well-behaved function, but a finite sum is still an approximation. Exactness depends on the function and the sampling method.

Why can a Riemann sum be negative?
When the sampled function heights are mostly below the x-axis, their signed rectangle areas are negative. This reflects signed area, not a negative geometric size.

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in a | Interval start a | - | 0 | 
in b | Interval end b | - | 4 | 
in n | Number of rectangles n | - | 4 | 0
list heights | Rectangle heights f(x_i) | - | 1, 2, 3, 4 | 
out sum | Left Riemann sum | function units | (b - a) / n * sum(heights) | 2
out dx | Rectangle width Δx | x units | (b - a) / n | 2