Riemann Sum Calculator

Riemann Sum Calculator

Enter equation, limits, number of rectangles, and select the type. The Riemann sum calculator computes the definite integrals and finds the sample points, with calculations shown.

Enter a Function:

W.R.T:

Lower Limit:

Upper Limit:

Number of rectangles:

Type:

The Riemann Sum Calculator with steps allows you to estimate a definite integral by summing sample points such as midpoints, trapezoids, right endpoints, and left endpoints using finite sums.

Keep reading to learn how to compute Riemann sums and the formulas involved.

What is a Riemann Sum?

In mathematics, a Riemann sum is a method to estimate the definite integral of a function over a given interval [a, b]. It was introduced by B. Riemann (1826–1866), a German mathematician. Riemann sums systematically approximate the area under a curve using a finite number of rectangles or shapes.

Riemann Sum Formula:

The Riemann sum S of a function f over an interval I with partition P is:

$$S = \sum_{i=1}^n f(x^*_i) \Delta x$$

Where:

$\Delta x = x_i - x_{i-1}$
$x_i^* \in [x_{i-1}, x_i]$ is a sample point in each subinterval

The choice of $x_i^*$ produces different Riemann sums (left, right, midpoint, or trapezoidal). For convenience, an online Riemann sum calculator can compute all approximations step by step instantly.

Methods of Riemann Sum

Divide the interval [a, b] into n subintervals, each of length:

$$\Delta x = \frac{b - a}{n}$$

The partition points are:

$$a, a + \Delta x, a + 2\Delta x, \dots, a + (n-1)\Delta x, b$$

Left Riemann Sum

The left Riemann sum estimates the integral using the left endpoint of each subinterval. The height of each rectangle is $f(a + i \Delta x)$ and the base is $\Delta x$. Summing these areas gives:

$$A_{Left} = \Delta x [ f(a) + f(a + \Delta x) + \dots + f(a + (n-1)\Delta x) ]$$

Right Riemann Sum

The right Riemann sum uses the right endpoint of each subinterval. The height of each rectangle is $f(a + i \Delta x)$, for $i = 1, \dots, n$, and the base is $\Delta x$:

$$A_{Right} = \Delta x [ f(a + \Delta x) + f(a + 2 \Delta x) + \dots + f(b) ]$$

How Riemann Sum Calculator Works

Input:

Enter the function to integrate, along with the upper and lower limits.
Specify the number of subintervals (rectangles) and select a sample point method (left, right, midpoint, trapezoid).
Click the “Calculate” button.

Output:

The calculator displays the standard form of the function and its limits.
Provides step-by-step calculations of the Riemann sum.
Optionally, outputs a table showing each subinterval, sample points, and rectangle areas.

This tool allows you to quickly organize results in a clear table using the Riemann sum table calculator, making estimation of definite integrals efficient and accurate.

Reference:

From Wikipedia: Riemann sum, Left Riemann sum, Right Riemann sum, Midpoint rule, Trapezoidal rule.