Goal
Left, right, and trapezoidal sums from evenly spaced heights.
Riemann Sums
Left, right, and trapezoidal sums from evenly spaced heights.
Key facts
- What it does
- Left, right, and trapezoidal sums from evenly spaced heights.
- Formula
n = m-1; h = (b-a)/n; left/right/trapezoid weightings of heights.- You enter
- Heights · Left endpoint a · Right endpoint b · Rule
- Worked example
- Left 1; right 5; trapezoid 3.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Heights · Left endpoint a · Right endpoint b · Rule
Method
n = m-1; h = (b-a)/n; left/right/trapezoid weightings of heights.
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Riemann Sums
Left, right, and trapezoidal sums from evenly spaced heights.
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Result
Enter your values above and choose Calculate to see the result here.
Calculation map
Follow the path from input to answer
01
Inputs (4)
- Heights Ready
- Left endpoint a Ready
- Right endpoint b Ready
- Rule Ready
02
Formula
n = m-1; h = (b-a)/n; left/right/trapezoid weightings of heights.
Bounded, transparent calculation
03
Result
- Calculate to preview the result.
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Formula, assumptions, and example
Formula: n = m-1; h = (b-a)/n; left/right/trapezoid weightings of heights.
Width comes from the interval split into height-count minus one parts. Left drops the last height, right drops the first, trapezoid halves the ends.
- At least 2 finite heights, evenly spaced over [a, b].
- Right endpoint b exceeds left endpoint a.
Worked example: Left 1; right 5; trapezoid 3.
Displayed input contract
- Heights
- Left endpoint a · minimum -1000000000000 · maximum 1000000000000
- Right endpoint b · minimum -1000000000000 · maximum 1000000000000
- Rule · 3 choices
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Riemann Sums for a real question
Left, right, and trapezoidal sums from evenly spaced heights. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes riemann sum, left right trapezoid, integral estimate. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Heights · Left endpoint a · Right endpoint b · Rule. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. At least 2 finite heights, evenly spaced over [a, b].
Need a wider view? Browse Math Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Riemann Sums
- Enter Heights — At least 2 comma-separated values, evenly spaced.
- Enter Left endpoint a.
- Enter Right endpoint b — Must exceed a.
- Enter Rule.
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
n = m-1; h = (b-a)/n; left/right/trapezoid weightings of heights.
Width comes from the interval split into height-count minus one parts. Left drops the last height, right drops the first, trapezoid halves the ends.
Worked example
Left 1; right 5; trapezoid 3.
Assumptions and limits
- At least 2 finite heights, evenly spaced over [a, b].
- Right endpoint b exceeds left endpoint a.
Context and background
The mathematical structure behind the tools
Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.
Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
A Riemann sum turns a finite row of sampled heights into a signed area estimate over an interval. This calculator does not receive a formula for f, draw a curve, or infer values between observations. It receives the text in the Heights field, the left endpoint a, the right endpoint b, and one of three rules: left endpoints, right endpoints, or trapezoids. It then treats the heights as evenly spaced samples, forms one interval for each gap between neighboring values, and reports the Riemann sum, the step width, and the number of intervals. That distinction matters. The output is an auditable calculation from supplied data, not a promise that the exact integral of an unknown function has been recovered. This guide explains the input grammar, the three weightings, worked calculations, display behavior, useful interpretations, and the situations in which the result should not be treated as more precise than the measurements behind it.
What a Riemann sum represents
Suppose a quantity has a value that changes along an x-axis. If the interval from a to b is divided into smaller pieces, a height at each sample location can be paired with the width of a piece. Height times width gives a rectangle-like contribution, and adding the contributions gives a numerical approximation to signed area. The word signed is important: a height below the horizontal axis contributes negatively, just as a positive height contributes positively. The calculator preserves that sign instead of replacing every height with an absolute value.
In textbook calculus, the heights often come from evaluating a known function at selected x-values. Here the heights are entered directly. They may be measurements, values exported from another model, or hand-entered samples. The calculator therefore answers a narrower and more transparent question: what sum results from applying the selected rule to these values on this interval? It cannot establish what the function did between the samples because no rule for those unsampled locations is part of the input.
The interval must have a genuine left-to-right direction. A positive b-a produces a positive step width, and the sign of the result then follows the supplied heights. If the data describe a rate, the result may have rate-units times x-units. If the data describe a signed profile, it is a weighted sum in the corresponding combined units. The page does not attach a physical unit automatically, so the person using it must document what the heights and endpoints mean.
- A sum combines supplied heights with a common horizontal width.
- Negative heights contribute negative signed area.
- The result is data-driven and does not identify a hidden function.
- The meaning of the units comes from the entered measurement context.
The four calculator fields
The Heights field is text rather than a single number. Enter a comma-separated or space-separated list of finite numeric values, such as 0, 1, 4 or 0 1 4. There must be at least two values because one value creates no interval. The values are ordered from the left side of the interval to the right side. The first height belongs at a and the last height belongs at b under the calculator's evenly spaced interpretation.
The Left endpoint a and Right endpoint b fields are numbers, not members of the height list. Each must be finite and between -1e12 and 1e12, and b must be greater than a. The endpoint values establish the horizontal domain and are used only to calculate the common width. They do not replace the first and last heights. A list can begin with a negative height even when a is positive, or end with a positive height when b is negative; the two kinds of values describe different axes.
The Rule field is a select choice with the exact values left, right, and trapezoid. Left endpoints use every height except the last. Right endpoints use every height except the first. Trapezoid uses half weight for both endpoint heights and full weight for interior heights. Selecting one rule does not alter the input samples or the interval count. It changes only how those same samples are weighted.
- Heights is a list of at least two values, in left-to-right order.
- a and b are finite endpoints with b greater than a.
- The available rules are left, right, and trapezoid.
- The endpoint fields define horizontal distance; heights define vertical values.
How list parsing and validation work
The handler first requires Heights to be a nonempty string. It trims surrounding whitespace, rejects text longer than 32,000 characters, and splits on one or more commas or whitespace characters. A comma at the beginning or end, or two commas with no value between them, is rejected explicitly. This catches a common spreadsheet or hand-editing mistake instead of silently shortening the data. Spaces may separate values, so a line break or several spaces is treated as another separator.
Each token must match a complete decimal-number grammar. Signed values, decimal points, and scientific notation are allowed, provided the token represents a finite JavaScript number. Text such as Infinity, NaN, hexadecimal notation, or a number with trailing nonnumeric characters is not accepted. The converted value must lie between -1e12 and 1e12. The parser does not evaluate expressions, infer missing entries, or treat an empty comma field as zero.
After conversion, the list may contain no more than 1,000 values. This is a count limit as well as a practical computation limit. The handler then checks that at least two values exist. These checks happen before the endpoint and rule calculations, so a malformed list cannot produce a partially meaningful sum. If a value came from an instrument export, inspect the delimiter, decimal convention, and missing-value markers before pasting it into the field.
- Comma and whitespace separators are accepted, but empty comma fields are not.
- Every token must be a complete finite decimal number.
- Each height is limited to -1e12 through 1e12.
- The input text is limited to 32,000 characters and 1,000 values.
- Expressions and missing-value placeholders are not evaluated.
Intervals and step width
Let m be the number of parsed heights. The handler defines n = m - 1. This is the number of intervals, not the number of samples. Three heights create two intervals, ten heights create nine intervals, and 1,000 heights create 999 intervals. Confusing m and n changes the width and produces a result that is scaled incorrectly, so the reported Intervals value is useful for checking the setup.
The common step width is h = (b-a)/n. Because b is required to exceed a and n is positive, h is positive. The implied sample locations are a, a+h, a+2h, and so on through b. The calculator never constructs those locations explicitly; it only needs their common spacing. This is why the list must represent equally spaced samples. If the original observations were taken at uneven x-values, one common h cannot represent their geometry without first resampling or using a different method.
The units of h are the units of the endpoints. If a and b are seconds, h is seconds. If heights are meters per second, a sum formed from height times h has meter units. The calculator displays Step width as a number without a unit label because the catalog does not know the endpoint unit. Keep the unit beside the result in a report or worksheet instead of assuming that the page has inferred it.
- m is the height count, while n = m - 1 is the interval count.
- The width is h = (b-a)/n.
- The implied sample locations are equally spaced from a through b.
- The user supplies the interpretation and units of the horizontal axis.
The left endpoint rule
For the left rule, each interval uses the height at its left boundary. With heights y0 through yn, the sum is h times (y0 + y1 +... + y(n-1)). The final height yn is not used because it sits at the right edge of the last interval. In code, this is the list with its last value removed, followed by a reduction and multiplication by h. The rule is a rectangle model whose top edge is determined by the earlier sample in each pair.
If the sampled values generally increase from left to right, the left rule tends to use shorter rectangles than the values near the right side of each interval, so it often sits below a smooth increasing curve. If the values generally decrease, the opposite tendency often occurs. Those are directional clues, not universal error guarantees. A curve can turn, oscillate, or contain measurement noise, and the left estimate can then be above or below a reference value.
The left rule still uses the full horizontal interval. Dropping the final height does not drop the final interval; it selects the height at that interval's left edge. This is a frequent off-by-one error when doing the calculation by hand. For a list of 0, 1, 4 on [0, 2], there are two intervals and the left heights are 0 and 1, not 0 and 1 and 4.
- Formula: left sum = h times the sum of all but the last height.
- Each rectangle uses the value at its interval's left edge.
- The final sample is intentionally excluded from the rectangle heights.
- Increasing or decreasing trends suggest direction but do not prove an error sign.
The right endpoint rule
For the right rule, each interval uses the height at its right boundary. The formula is h times (y1 + y2 +... + yn), so the first height y0 is omitted and every later height is used. This is the counterpart to the left rule. The interval count and step width do not change; only the selected rectangle heights change.
On a smooth increasing pattern, right rectangles often lie above the curve while left rectangles often lie below it. On a decreasing pattern, their typical ordering reverses. Comparing the two can therefore provide a rough sense of how much the sampling choice matters. A wide gap between them is a warning that the interval is coarse or the values change substantially between neighboring samples. A narrow gap is encouraging, but it is not proof that unsampled behavior is harmless.
The right rule is especially natural when each value is a reading recorded at the end of a period. For example, if a row of readings is explicitly defined as the amount available at the end of each interval, using those readings as right-edge values may match the data collection convention. The calculator does not know that convention on its own, so state it when reporting the result.
- Formula: right sum = h times the sum of all but the first height.
- Each rectangle uses the value at its interval's right edge.
- The first sample is excluded from the right rectangle heights.
- A left-versus-right comparison reveals sensitivity to endpoint selection.
The trapezoid rule
The trapezoid rule joins neighboring samples with a straight segment and uses the area of the resulting trapezoid on each interval. When the same h is used throughout, the combined formula is h times (y0/2 + y1 + y2 +... + y(n-1) + yn/2). Interior samples belong to two neighboring trapezoids, so their combined weight is one full height. The first and last samples belong to only one trapezoid, so each receives half weight.
The handler computes the interior total from the values between the endpoints, then adds half of the first and half of the last. It does not average the left and right final numbers as a separate shortcut, although with evenly spaced samples the algebraic result is equivalent to the average of those two sums. The direct endpoint-weight formula makes the treatment of each sample explicit and keeps the rule understandable when negative values are present.
A trapezoid estimate assumes a linear connection between adjacent sampled values. It can be more representative than a one-sided rectangle when the sample values change smoothly, because both ends of every interval influence the shape. It can still miss strong curvature, sharp peaks, discontinuities, or narrow features between samples. A visually smooth sequence is evidence for a plausible interpolation, not a guarantee about the underlying process.
- Formula: trapezoid sum = h times half endpoints plus full interior heights.
- The first and last samples have weight one-half.
- Interior samples have full combined weight.
- The rule models each gap with a straight-line connection.
Worked example with the default values
Use Heights = 0, 1, 4, a = 0, and b = 2. There are m = 3 samples, so n = 3 - 1 = 2 intervals. The common width is h = (2 - 0)/2 = 1. The implied x locations are 0, 1, and 2. The data say only that the sampled heights at those locations are 0, 1, and 4; they do not specify a unique curve between them.
For the left rule, use the first two heights: 1 times (0 + 1) = 1. For the right rule, use the last two heights: 1 times (1 + 4) = 5. For the trapezoid rule, use 1 times (0/2 + 1 + 4/2) = 1 times (0 + 1 + 2) = 3. These are three valid estimates from the same inputs because the rules answer slightly different sampling questions.
The contrast is informative. The sequence rises sharply at the final sample, so the left rule does not see the height 4 as a rectangle top while the right rule does. The trapezoid rule gives that endpoint half weight, placing its result midway between the two rectangle estimates in this case. If the heights represented a nonnegative rate over two time units, the three numbers would be three competing accumulated amounts in rate-time units.
- Height count m = 3 and interval count n = 2.
- Step width h = 1.
- Left sum = 1, right sum = 5, and trapezoid sum = 3.
- The spread reflects the final sample's strong increase.
Worked example with signed values
Consider Heights = 3, 1, -2 on the interval from a = -1 to b = 5. Again m = 3 and n = 2, but now h = (5 - (-1))/2 = 3. The implied sample positions are -1, 2, and 5. The negative final height is not an error by itself; it may represent a quantity below a reference level, a negative rate, or a signed measurement.
The left calculation is 3 times (3 + 1) = 12. The right calculation is 3 times (1 + (-2)) = -3. The trapezoid calculation is 3 times (3/2 + 1 + (-2)/2) = 3 times (1.5 + 1 - 1) = 4.5. Notice that the right result is negative even though two of the three samples are positive. It uses the final two values, whose sum is negative, and the sign is meaningful under the signed-area convention.
This example also shows why taking absolute values would change the question. Replacing -2 by 2 would make the right sum positive and would erase information about direction or position relative to the reference level. If a physical application wants total magnitude instead of net signed accumulation, that is a separate analysis choice. First calculate the signed result faithfully, then decide whether an absolute-value or positive-part analysis is justified by the application.
- Signed heights remain signed in every rule.
- For 3, 1, -2 on [-1, 5], h = 3.
- The three results are 12, -3, and 4.5 for left, right, and trapezoid.
- Do not take absolute values unless the application explicitly asks for magnitude.
Comparing rules and reading disagreement
The three rules use the same data but encode different assumptions about where a representative height lies inside each interval. Left and right are one-sided rectangle choices. Trapezoid uses both neighboring endpoints and is often a balanced choice for a smoothly changing series. Reporting more than one rule can be useful when the data are sparse because the spread gives a simple sensitivity check rather than hiding a choice behind one number.
For an increasing sequence, a large right-minus-left difference usually points to a large net change across the samples. Algebraically, the difference is h times the final height minus the first height when the list is evenly spaced, because the interior terms cancel. This identity explains why endpoint contrast, not every individual fluctuation, controls the total gap between the two rectangle rules. It is a diagnostic, not an error bound for an arbitrary unknown process.
The trapezoid result is the average of the left and right results under this equal-spacing setup. That relationship is useful for checking arithmetic: calculate both rectangle sums independently, average them, and compare with the displayed trapezoid value. If the numbers disagree beyond ordinary floating-point display effects, inspect the list order, endpoint values, interval count, and selected mode.
- Rule disagreement is a sensitivity signal, not automatically a bug.
- Right minus left equals h times the last height minus the first for equal spacing.
- The trapezoid value equals the average of left and right under this contract.
- A wide spread often calls for more samples or a better-defined model.
Accuracy, convergence, and sample design
A Riemann sum becomes more informative when the sample spacing is small relative to the changes in the quantity being studied. If a function is smooth and the samples become more closely spaced, rectangle and trapezoid approximations can approach the corresponding integral under the usual calculus conditions. In this page, however, the user supplies a finite list only once. The calculator cannot test a sequence of refinements or prove that the values came from a smooth function.
A practical refinement exercise is to collect or compute more samples on the same interval, then compare results from the old and new lists. Keep the endpoint meaning consistent and preserve the same rule. If the values are measurements, consider noise, instrument resolution, and whether adding samples changes the measurement process. If the values come from a model, check whether the new points are actually evaluated at equal x-spacings rather than merely appended in time order.
Sharp peaks and narrow valleys are especially dangerous. A list can look stable at its sample locations while missing a large feature between them. A discontinuity can be hidden by two finite neighboring values, and rapid oscillation can make left, right, and trapezoid sums accidentally agree while the true accumulated behavior differs. Agreement among rules is useful evidence, but it is not a certificate of accuracy without information about the process between samples.
- More samples can improve an estimate only when they represent the same interval and spacing correctly.
- Smoothness supports convergence arguments; arbitrary measured lists do not prove them.
- Narrow features can be missed by every supplied sample.
- Comparing refinements is stronger evidence than relying on one displayed total.
Validation edge cases
Exactly two heights are valid and create one interval. With heights y0 and y1, the left result is h times y0, the right result is h times y1, and the trapezoid result is h times (y0 + y1)/2. This smallest case is useful for testing endpoint conventions. A one-value list is rejected because there is no gap over which to apply a width, even if the user might imagine a constant function.
The endpoints may be negative, zero, or positive, as long as both are finite, within the specified bounds, and ordered b greater than a. A very small positive interval is allowed when its endpoints are distinct. The mode must be one of the three select values; arbitrary text such as average or midpoint is rejected. The handler does not coerce a numeric-looking endpoint string into a number, because the calculation contract expects numeric field values.
The list bounds are applied to individual heights, not to the total. A valid list can still produce a large sum, especially with many entries or a wide interval. The output helper rejects a nonfinite numeric result and asks for smaller inputs if multiplication or accumulation leaves the representable finite-number range. That is preferable to returning Infinity as though it were a meaningful area.
- Two heights are the minimum valid list and create one interval.
- b equal to a is rejected because the width would be zero.
- Numeric bounds apply to each height and each endpoint independently.
- A finite input list can still produce an unrepresentable nonfinite result.
Numerical and display behavior
The handler calculates with JavaScript finite numbers. It normalizes negative zero in its general numeric output helper, so a result that is mathematically zero is not intentionally displayed as a signed negative zero. Intermediate summation follows the order of the parsed list. With very large positive and negative values, ordinary floating-point cancellation can make the last digits sensitive to order and scale even though the input values pass validation.
Riemann sum and Step width are returned as numeric results with precision metadata of 6. Intervals is numeric with precision 0. The renderer uses that metadata for presentation, while the stored result remains a number rather than a text approximation. The calculation steps use a compact formatter that normally rounds to six decimal places, removes unnecessary trailing zeros, and keeps a very small nonzero value in a direct string form rather than forcing it to display as zero.
The displayed six-place result should not be read as six guaranteed significant figures. If the heights were measured to only two meaningful digits, the measurement uncertainty may dominate the last displayed places. Conversely, if a very small step width matters, a rounded display can hide its scale even though the numeric value remains available to the renderer. Record the raw inputs, units, rule, and displayed precision when the result is used in an audit or report.
- The numeric result is retained separately from its formatted display.
- Riemann sum and step width use precision 6; intervals uses precision 0.
- Negative zero is normalized in output values.
- Floating-point cancellation and input measurement precision can limit meaningful digits.
Practical uses
The calculator is useful for a first-pass accumulation from a table. A distance estimate can use sampled speed values over equal time intervals. A total load can use sampled intensity over equal positions. A signed balance can use positive and negative deviations from a baseline. In each case, the result is most defensible when the list is accompanied by a clear statement of what one height measures and what a and b measure.
It is also useful for teaching and review. Entering the same list under all three modes makes endpoint weighting visible. Repeating the calculation after adding samples illustrates why interval count is one less than sample count. A student can compute the formula by hand, compare each intermediate sum with the page, and use the result as a check without treating the calculator as a replacement for understanding the sampling convention.
For an engineering, laboratory, or business worksheet, preserve the source table separately. The page accepts a compact list and does not retain provenance such as timestamps, sensor identifiers, calibration state, missing-value policy, or the reason a sample was taken. Those details may be more important than the final decimal when another person needs to reproduce the estimate or decide whether the data were suitable for integration.
- Use it to accumulate sampled rates, profiles, loads, or deviations over equal spacing.
- Compare modes to expose endpoint-weighting sensitivity.
- Keep the original table and measurement metadata outside the calculator.
- State the physical units and sign convention beside the result.
Model limitations
The strongest limitation is that the page has no hidden function. It cannot determine whether a curve is increasing between samples, whether a maximum was missed, or whether a straight-line connection is physically reasonable. It also cannot estimate an error without additional assumptions about smoothness, derivatives, curvature, or measurement uncertainty. Calling the output an integral is appropriate only when the entered samples and the surrounding mathematical model justify that interpretation.
The common-width rule excludes irregularly spaced observations. If the x locations are not equally spaced, do not paste their heights and let the calculator pretend that the gaps are equal. Re-express the data on an equal grid if that is scientifically appropriate, or use a method that accepts the actual x locations. Similarly, the page does not interpolate missing values, smooth noise, remove outliers, or convert units.
The numeric bounds are safeguards, not claims that every value in the range is physically meaningful. A list of 1,000 extreme heights may be legal input but still be a poor model of a real process. The result does not include uncertainty intervals, confidence statements, conservation checks, domain-specific constraints, or a decision recommendation. It is a transparent arithmetic component that must be paired with domain review when the conclusion matters.
- No function, interpolation, error estimate, or uncertainty model is inferred.
- Unequal x-spacing is outside the contract.
- The handler does not smooth, filter, resample, or convert units.
- A bounded arithmetic result is not automatically a physically valid conclusion.
A repeatable reporting workflow
Start by naming the quantity represented by each height and the horizontal quantity represented by the endpoints. Confirm that the samples are in chronological or spatial order and are equally spaced. Then check that the list has no missing markers, accidental blank comma fields, or values in the wrong unit. Write down m, n, and h before choosing a rule. These simple notes make a later review much easier than a lone final number.
Run the rule that matches the measurement convention, and when the data are sparse run the other two as sensitivity comparisons. Reproduce the endpoint weighting on paper or in a separate spreadsheet. Check the sign using a rough sketch or the direction of the values. If a positive rate produces a negative accumulation, inspect the list and endpoint order before interpreting the output as surprising behavior.
Finally, report the interval, the sample list or its source, the chosen rule, the step width, the returned sum, and the display precision. Describe the result as an estimate based on evenly spaced supplied heights. If the estimate will support a material decision, add an independent check using more samples, a known function, or a domain-specific integration method. This workflow keeps the useful simplicity of the calculator while making its assumptions visible.
- Document what each height and endpoint means before calculating.
- Verify ordering, equal spacing, delimiters, units, and signs.
- Use other rules or denser samples as sensitivity checks.
- Report the method and assumptions, not only the final decimal.
Frequently asked questions
What is the Riemann Sums?
Left, right, and trapezoidal sums from evenly spaced heights.
What is the formula for the Riemann Sums?
n = m-1; h = (b-a)/n; left/right/trapezoid weightings of heights. Width comes from the interval split into height-count minus one parts. Left drops the last height, right drops the first, trapezoid halves the ends.
What do I need to use this calculator?
Enter Heights, Left endpoint a, Right endpoint b, Rule, then choose Calculate.
What are the limits of this calculator?
At least 2 finite heights, evenly spaced over [a, b]. Right endpoint b exceeds left endpoint a.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
Use this calculator as part of a bigger plan
These WorldCalculate collections connect this tool with related questions while keeping each calculation separate and transparent.