Riemann Sum Calculator
Approximate definite integrals using left, right, midpoint, and trapezoidal Riemann sums.
Visualization method
All calculations run live in your browser. Supports standard functions: sin, cos, tan, sqrt, exp, log, pi, e. Trapezoid rule is generally more accurate for smooth functions.
What Is the Riemann Sum Calculator?
A Riemann sum approximates a definite integral by dividing the area under a curve into n rectangles (or trapezoids) and summing their areas. This calculator computes all four methods simultaneously with an SVG visualization showing the rectangles, a comparison table, and optional error analysis against a known exact value.
- ›Left/Right sums can over- or under-estimate depending on function behavior
- ›Midpoint rule is generally more accurate than left/right for smooth functions
- ›Trapezoidal rule uses a linear interpolation between endpoints, O(Δx²) accuracy
- ›All four methods converge to the exact integral as n → ∞
Formula
Four Riemann Sum Methods
Left sum
Σ f(xᵢ) × Δx (i = 0 to n−1)
Right sum
Σ f(xᵢ₊₁) × Δx (i = 0 to n−1)
Midpoint
Σ f((xᵢ+xᵢ₊₁)/2) × Δx
Trapezoid
Σ (f(xᵢ)+f(xᵢ₊₁))/2 × Δx
Step size
Δx = (b − a) / n
Exact integral
∫ₐᵇ f(x) dx = lim(n→∞) Riemann sum
How to Use
- 1Select a preset function (x², sin(x), √x…) or type your own f(x)
- 2Enter lower bound a and upper bound b (supports "pi" and "e")
- 3Set the number of subdivisions n (1–10,000; more n = more accuracy)
- 4Optionally enter the exact value to see the percentage error for each method
- 5Choose a visualization method, the chart shows that method's rectangles
- 6Click Calculate to see all four sums and the error analysis
Example Calculation
Approximate ∫₀¹ x² dx with n=4:
Left: [f(0)+f(0.25)+f(0.5)+f(0.75)]×0.25
= [0+0.0625+0.25+0.5625]×0.25 = 0.21875
Right: [f(0.25)+f(0.5)+f(0.75)+f(1)]×0.25
= [0.0625+0.25+0.5625+1]×0.25 = 0.46875
Midpoint: [f(0.125)+f(0.375)+f(0.625)+f(0.875)]×0.25
= 0.328125
Trapezoid: (Left+Right)/2 = 0.34375
Exact = 1/3 = 0.333333...
Midpoint error: 1.5% · Trapezoid error: 3.1%
Why midpoint beats trapezoid for x²
For convex functions (concave up), the trapezoidal rule overestimates. The midpoint rule underestimates by roughly half as much, so midpoint has about half the error of trapezoid for smooth functions.
Understanding Riemann Sum
Method Accuracy Comparison
| Method | Error Order | n=10 typical error | Best for |
|---|---|---|---|
| Left / Right | O(Δx) | ~5–10% | Teaching, worst-case bounds |
| Midpoint | O(Δx²) | ~0.5–1% | Smooth functions, general use |
| Trapezoidal | O(Δx²) | ~0.5–2% | Piecewise / tabulated data |
| Simpson's 1/3 | O(Δx⁴) | ~0.001% | Smooth functions, high accuracy |
Frequently Asked Questions
What is a Riemann sum?
Named after Bernhard Riemann, the Riemann sum is both a practical numerical method and the rigorous definition of the Riemann integral in calculus.
- ›Each rectangle has width Δx = (b−a)/n and height f(x*ᵢ) at a sample point
- ›Left: sample point = left endpoint; Right: right endpoint; Midpoint: midpoint
- ›Trapezoid: average height of left and right endpoints
- ›As n → ∞, all four methods converge to the exact definite integral
Which method is most accurate?
Accuracy depends on the function's smoothness. For well-behaved functions, the hierarchy is typically:
- ›Least accurate: Left or Right sum, O(Δx) error (halves when n doubles)
- ›Better: Midpoint and Trapezoidal, O(Δx²) error (quarters when n doubles)
- ›Best basic method: Midpoint slightly edges trapezoidal for convex functions
- ›Even better: Simpson's 1/3 rule, O(Δx⁴) (uses midpoint+trapezoid combination)
How does increasing n affect accuracy?
This is the concept of convergence rate. Higher-order methods achieve the same accuracy with fewer subdivisions.
- ›Left sum with n=100: error ~1%; n=10,000: error ~0.01%
- ›Midpoint with n=100: error ~0.01%; n=1,000: error ~0.0001%
- ›n=1,000 midpoint ≈ n=100,000 left sum for the same accuracy
- ›Enter an exact value in the tool to see the actual error percentage
What functions can I enter?
The expression evaluator handles standard mathematical notation. Use * for multiplication and ^ for powers.
- ›Polynomials: x^3 - 2*x + 1
- ›Trig: sin(x)*cos(x) or sin(x)^2
- ›Exponentials: exp(-x^2) (Gaussian), exp(x)/x
- ›Combinations: sqrt(1-x^2) (semicircle), x*log(x)
Why does the trapezoidal rule overestimate for convex functions?
The direction of error depends on function concavity, which is determined by the second derivative.
- ›Convex (f′′>0, concave up): trapezoid overestimates, midpoint underestimates
- ›Concave (f′′<0, concave down): trapezoid underestimates, midpoint overestimates
- ›Linear functions (f′′=0): all methods give the exact answer
- ›Simpson's rule cancels these errors: (2×Midpoint + Trapezoid) / 3
What is Simpson's rule and how does it relate?
Simpson's rule achieves fourth-order accuracy by combining midpoint and trapezoid sums in a 2:1 ratio, eliminating the leading error term from both methods.
- ›Simpson's rule = (2 × Midpoint + Trapezoid) / 3
- ›Error order: O(Δx⁴), error quarters when Δx is halved
- ›For ∫₀¹ x² dx: Simpson = (2×0.333+0.344)/3 ≈ 0.3333, exact!
- ›The Integral Calculator on this site uses Simpson's rule
What is the visualization showing?
The visualization is pedagogically valuable, you can see how the rectangles over- or under-shoot the curve and how increasing n fills the gaps.
- ›Blue rectangles show the Riemann approximation area
- ›The dark blue curve shows the actual function f(x)
- ›Left sum: rectangles touch left endpoint of each interval
- ›Midpoint sum: rectangles center on interval midpoint