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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:

Upper Limit:

Lower Limit:

Number of rectangles:

Type:

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Riemann Sum Calculator with Steps

The Riemann Sum Calculator allows you to estimate a definite integral by summing sample points such as midpoints, trapezoids, right endpoints, and left endpoints. It provides step-by-step calculations for better understanding of each method.

What is a Riemann Sum?

In mathematics, a Riemann sum is a method for approximating the definite integral of a function over an interval [a, b]. Introduced by B. Riemann (1826–1866), it estimates 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 $x_i^*$ gives different Riemann sums: left, right, midpoint, or trapezoidal.

Methods of Riemann Sum

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

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

Partition points:

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

Left Riemann Sum

Uses the left endpoint of each subinterval. Height = $f(a + i \Delta x)$, base = $\Delta x$:

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

Right Riemann Sum

Uses the right endpoint of each subinterval. Height = $f(a + i \Delta x)$, for $i = 1, \dots, n$, base = $\Delta x$:

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

Midpoint Riemann Sum

Uses the midpoint of each subinterval as the sample point:

$$A_{Mid} = \Delta x \sum_{i=1}^{n} f\Big(a + (i - 0.5)\Delta x\Big)$$

Trapezoidal Rule

Approximates the area using trapezoids instead of rectangles:

$$A_{Trap} = \frac{\Delta x}{2} \big[f(a) + 2\sum_{i=1}^{n-1} f(a + i \Delta x) + f(b)\big]$$

How the Riemann Sum Calculator Works

Input:

Enter the function to integrate and the interval limits [a, b].
Specify the number of subintervals (n) and the sample point method (left, right, midpoint, trapezoid).
Click Calculate.

Output:

Displays the function and its limits.
Provides step-by-step calculations of the Riemann sum.
Optionally shows a table with subintervals, sample points, and rectangle/trapezoid areas.

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

References

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

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