Computed Result
The calculator places the result here, above the form, after submission.
Plotly Graph
Interval Breakdown
| # | xi | xi+1 | Width | Sample x | f(sample) | Interval Area |
|---|
Calculator Inputs
Supported functions include sin(x), cos(x), exp(x), log(x), sqrt(x), abs(x), and powers like x^2.
Example Data Table
Example using f(x)=x² on [0,2] with midpoint method and 4 equal subintervals.
| # | Interval | Midpoint | f(midpoint) | Width | Area |
|---|---|---|---|---|---|
| 1 | [0.00, 0.50] | 0.25 | 0.0625 | 0.50 | 0.03125 |
| 2 | [0.50, 1.00] | 0.75 | 0.5625 | 0.50 | 0.28125 |
| 3 | [1.00, 1.50] | 1.25 | 1.5625 | 0.50 | 0.78125 |
| 4 | [1.50, 2.00] | 1.75 | 3.0625 | 0.50 | 1.53125 |
| Total Midpoint Riemann Sum | 2.62500 | ||||
Formula Used
General Riemann Sum
For a partition of [a,b] into subintervals, the approximation is:
S = Σ f(xi*) Δxi
Left Sum: sample point is the left endpoint.
Right Sum: sample point is the right endpoint.
Midpoint Sum: sample point is the interval midpoint.
Custom Sum: sample point is xi + tΔxi, where 0 ≤ t ≤ 1.
Trapezoidal Sum: ((f(xi) + f(xi+1)) / 2) Δxi
Lower Sum: uses the minimum function value on each interval.
Upper Sum: uses the maximum function value on each interval.
The page also computes a high-resolution numerical reference integral. That estimate helps you compare approximation error and observe convergence as partitions become finer.
How to Use This Calculator
- Enter the function using x, such as
sin(x),x^3 - 2*x, orexp(-x^2). - Choose an approximation method that matches your study goal.
- Set the interval bounds, then choose a number of subintervals.
- Use uniform partitions for equal widths, or enter custom breakpoints for irregular partitions.
- For the custom method, enter a ratio between 0 and 1 to place the sample point inside each interval.
- Press the calculate button to show the result above the form.
- Review the graph, metrics, and interval table, then export CSV or PDF if needed.
Helpful Notes
Irregular partitions supported Reference integral comparison Lower and upper sums Plotly visualization CSV and PDF exportSome expressions may be undefined on parts of the interval. If the function contains discontinuities, vertical asymptotes, or invalid domains, refine the interval and inspect the graph carefully.
FAQs
1) What does a Riemann sum estimate?
A Riemann sum estimates the signed area under a curve over an interval. It does this by splitting the interval into smaller pieces and adding rectangle or trapezoid areas.
2) When is the midpoint method better than left or right?
The midpoint method often gives better accuracy for smooth functions because its sample point sits inside each interval. It frequently reduces bias that appears with left or right endpoint sampling.
3) What is the difference between lower and upper sums?
A lower sum uses the smallest function value on each subinterval. An upper sum uses the largest function value. Together, they help bracket the true integral for many bounded functions.
4) Why does increasing subintervals improve accuracy?
More subintervals usually mean narrower pieces, so the approximation follows the curve more closely. For well-behaved functions, the sum tends to converge toward the definite integral.
5) Can I use uneven partitions?
Yes. Uneven partitions are useful when a function changes rapidly in certain regions. You can enter custom breakpoints to make some intervals finer than others.
6) Does the calculator handle negative function values?
Yes. Areas below the x-axis contribute negative signed area. The reported sum reflects that sign, which is consistent with the meaning of definite integrals.
7) What expressions can I enter for f(x)?
You can enter algebraic, trigonometric, logarithmic, and exponential expressions supported by the parser, such as x^2, sin(x), log(x+2), and exp(-x).
8) Why compare the sum with a reference integral?
A reference integral helps you gauge approximation quality. Comparing both values reveals error size and direction, which is helpful when studying convergence or testing different methods.