Riemann Sum Calculator
Approximate definite integrals using Left Endpoint ($L_n$), Right Endpoint ($R_n$), Midpoint ($M_n$), Trapezoidal ($T_n$), and Simpson's ($S_n$) numerical integration with interactive rectangle area diagrams.
Area Approximation Partition Plot
∑ f(x*) ΔxStep-by-Step Riemann Sum Partition Summation Formula: Δx = (b − a) / n
How to Calculate a Riemann Sum
To calculate a Riemann sum for a function f(x) on interval [a, b] with n subintervals, first find the step width Δx = (b - a) / n. For a Left sum, evaluate f(x) at x₀, x₁, ..., x_{n-1} and multiply the sum by Δx. For a Right sum, evaluate at x₁, x₂, ..., x_n. For a Midpoint sum, evaluate at interval centers (x_{i-1} + x_i)/2. For the Trapezoidal rule, average the left and right endpoints with doubled interior terms: T_n = (Δx/2) [f(x₀) + 2f(x₁) + ... + f(x_n)].
Anatomy of Riemann Sums & The Definite Integral Definition
Named after the German mathematician Bernhard Riemann (1826–1866), a Riemann sum is the rigorous foundational mechanism used to define the definite integral in calculus.
The continuous region bounded by $y = f(x)$, the $x$-axis, and vertical lines $x = a$ and $x = b$ is sliced into $n$ vertical strips of width $\Delta x = \frac{b - a}{n}$. Each strip's area is approximated by a geometric shape whose height is sampled at a point $x_i^*$ within the strip:
The uniform base width of each subinterval slice.
The vertical height evaluated at left, right, or midpoint coordinates.
The summation of all $n$ rectangular or trapezoidal slices.
Left and Right Endpoint Approximation Formulas
Evaluates function at the starting left boundary of each slice ($x_0 = a$).
Evaluates function at the ending right boundary of each slice ($x_n = b$).
The Midpoint Rule (M_n) & Reduced Truncation Error
Rather than choosing an edge, the Midpoint Rule samples $f(x)$ at the exact center of each interval: $\bar{x}_i = \frac{x_{i-1} + x_i}{2}$.
Because the rectangle cuts through the curve near the middle, the triangular area underestimated on one side roughly cancels the overestimated area on the other side, achieving $O(1/n^2)$ convergence.
The Trapezoidal Rule (T_n) for Linear Segment Interpolation
The Trapezoidal Rule connects consecutive curve points with straight secant chords:
Note that interior vertices appear twice because adjacent trapezoids share inner boundaries, while the outer endpoints ($x_0$ and $x_n$) appear only once.
Simpson's Rule (S_n) for Parabolic Segment Approximations
Simpson's Rule fits second-degree quadratic polynomials through consecutive triplets of points:
Pattern of coefficients: $1, 4, 2, 4, 2, \dots, 4, 1$. (Requires an even number of subintervals $n$). It is exact for all cubic polynomials ($\deg \le 3$).
Convergence as n Approaches Infinity (The Fundamental Theorem)
The definition of the definite integral as the limit of a Riemann sum:
Step-by-Step Worked Examples (Polynomial and Trigonometric Integrals)
Approximate ∫₀² x² dx using Trapezoidal Rule with n = 4.
1. Δx = (2 − 0)/4 = 0.5.
2. Points: x₀=0, x₁=0.5, x₂=1.0, x₃=1.5, x₄=2.0.
3. f(x) values: f(0)=0, f(0.5)=0.25, f(1)=1, f(1.5)=2.25, f(2)=4.
4. T₄ = (0.5 / 2) [0 + 2(0.25) + 2(1) + 2(2.25) + 4] = 0.25 [0 + 0.5 + 2 + 4.5 + 4] = 0.25(11) = 2.7500.
Comparison: Exact = 8/3 ≈ 2.6667 (Error = +3.125%).
Common Pitfalls & Overestimation vs Underestimation Rules
In a Left Riemann sum with $n$ rectangles, do not include the upper bound $f(b)$; the evaluation stops at $x_{n-1}$. In a Right sum, start at $x_1$ and stop at $x_n = b$.
For concave-up curves ($f''(x) > 0$), the Trapezoidal rule always overestimates the true integral because secant lines lie above the curve. For concave-down curves ($f''(x) < 0$), it underestimates.
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