Riemann Sum Approximator | Left, Right, Midpoint & Trapezoid Calculator

RIEMANN SUM APPROXIMATOR

Left, Right, Midpoint & Trapezoidal Area Approximations

Calculus & Analysis

Integrand $f(x)$:

Lower Bound $a$:

Upper Bound $b$:

Number of Subintervals ($n$):

Riemann Sum & Area Approximations ($\Delta x = \text{0.5000}$)

Left Riemann Sum ($L_n$) 1.7500

Right Riemann Sum ($R_n$) 3.7500

Midpoint Rule ($M_n$) 2.6250

Trapezoidal Rule ($T_n$) 2.7500

Exact Definite Integral ($\int_a^b f(x) dx$) 2.6667

Step-by-Step Numerical Derivation

Riemann Sum Approximator: Left, Right, Midpoint & Trapezoidal Rules

A Riemann Sum is a fundamental technique in real analysis and numerical calculus for approximating the definite integral (area under a curve) of a function $f(x)$ over a bounded interval $[a, b]$. By partitioning the domain into $n$ equal subintervals of width $\Delta x = \frac{b - a}{n}$, the region is divided into rectangular or trapezoidal slices whose areas are summed.

1. Summary of Riemann Sum Approximations

Method	Sampling Point ($x_i^*$)	Summation Formula	Behavior on Monotonic Functions
Left Riemann Sum ($L_n$)	Left endpoint: $x_i^* = x_{i-1}$	$$L_n = \Delta x \sum_{i=0}^{n-1} f(x_i)$$	Underestimates increasing functions; overestimates decreasing functions.
Right Riemann Sum ($R_n$)	Right endpoint: $x_i^* = x_i$	$$R_n = \Delta x \sum_{i=1}^{n} f(x_i)$$	Overestimates increasing functions; underestimates decreasing functions.
Midpoint Rule ($M_n$)	Subinterval midpoint: $x_i^* = \frac{x_{i-1} + x_i}{2}$	$$M_n = \Delta x \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right)$$	Higher precision; cancellation of errors occurs across subintervals.
Trapezoidal Rule ($T_n$)	Average of endpoints	$$T_n = \frac{\Delta x}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$$	Connects points linearly; overestimates concave up curves, underestimates concave down curves.

2. Error Bounds & Convergence

As the number of subintervals $n \to \infty$ (and $\Delta x \to 0$), all valid Riemann sums converge to the exact value of the definite integral under Riemann integrability:

$$\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x = \int_{a}^{b} f(x) \, dx$$

Trapezoidal Error Bound: $$|E_T| \le \frac{K_2 (b - a)^3}{12 n^2} \quad \text{where } K_2 \ge |f''(x)|$$
Midpoint Error Bound: $$|E_M| \le \frac{K_2 (b - a)^3}{24 n^2}$$

Frequently Asked Questions

Why is the Midpoint Rule usually more accurate than the Trapezoidal Rule?

Although the Trapezoidal Rule uses two endpoints per slice, its error depends directly on linear secant lines. The Midpoint Rule uses tangent slopes at subinterval centers, leading to error cancellation where overestimation on one side of the midpoint balances underestimation on the other.

How does $n$ affect approximation accuracy?

Doubling $n$ halves $\Delta x$. For Left and Right sums, the error decreases linearly $\mathcal{O}(1/n)$. For Midpoint and Trapezoidal rules, doubling $n$ reduces error by a factor of 4 ($\mathcal{O}(1/n^2)$).

Back to Mathematics Tools

Found a Mistake or Have a Suggestion?

The AISkillsUp Academy team is always looking to improve our engines. If you spotted an error, have a query, or want to request a new feature for this specific tool, let us know! Drop a comment below, visit our contact page, or email us directly. We will investigate and update it immediately.

Riemann Sum Approximator | Left, Right, Midpoint & Trapezoid Calculator