Riemann Sum Calculator
Riemann Sum Calculator computes definite integrals numerically by approximating area under curves using left, right, midpoint, and trapezoidal partition methods.
Enter your function here. Example:x^2 or sin(x)
🔢 Button Pad
📌 Examples
Result
What is Riemann Sum Calculator
A Riemann Sum Calculator is a numerical tool that approximates the definite integral of a function over an interval by partitioning the domain into subintervals and summing the areas of simple geometric shapes (rectangles or trapezoids). It is a practical way to see how the integral arises as the limit of finite sums and is useful when an antiderivative is difficult or impossible to obtain symbolically. The calculator supports different sample points and refinement of partitions to improve accuracy.
About the Riemann Sum Calculator
This calculator provides several standard Riemann approximations: left Riemann sum, right Riemann sum, midpoint rule, and trapezoidal rule. It displays the sum formula, computes numeric results for a chosen number of subintervals \(n\), and optionally shows the sequence of sums as \(n\) increases so users can observe convergence to the definite integral
\[ \text{As } \max_i\Delta x_i\to 0,\quad \sum_{i=1}^{n} f(\xi_i)\Delta x_i \to \int_a^b f(x)\,dx. \]
How to Use this Riemann Sum Calculator
- Enter the integrand \(f(x)\) and the integration interval \([a,b]\).
- Choose the number of subintervals \(n\) (larger \(n\) yields higher accuracy).
- Select the approximation type: left, right, midpoint, or trapezoidal.
- Click Calculate to receive the numeric sum and an optional step-by-step breakdown showing sample points and area contributions.
- Optionally run a convergence test by increasing \(n\) and observing the sums approaching the integral value.
Example
Approximate \(\int_0^1 x^2\,dx\) with \(n=4\) using the midpoint rule. Partition points: \(0,\,0.25,\,0.5,\,0.75,\,1\). Midpoints: \(0.125,0.375,0.625,0.875\).
The Riemann Sum Calculator provides the numeric approximation, shows how each rectangle or trapezoid contributes to the total, and helps users connect geometric intuition with the formal limit definition of the definite integral. It is an excellent learning and checking tool for calculus students, engineers, and anyone needing reliable numerical integrals.