Limit of Riemann Sum to Definite Integral Calculator Online

This calculator transforms the concept of Riemann sums into definite integrals through the limit process. Ideal for both students and professionals, it offers a straightforward way to compute areas under curves without delving into the manual intricacies of calculus. By inputting a function, along with the interval limits, the calculator provides an accurate definite integral, saving time and enhancing understanding.

Formula

To comprehend the calculator’s operations, let’s dive into the underlying formula:

∫ₐᵇ f(x) dx = lim_(n→∞) Σ(f(x_i) * Δx)

where:

• Σ denotes summation (sum over i from 1 to n),
• f(x_i) is the function evaluated at some point x_i within the subinterval,
• Δx is the width of each subinterval, calculated as (b – a) / n.

Steps to Use the Formula:

1. Divide the interval [a, b] into n subintervals.
2. Choose a sampling strategy (left, right, midpoint, trapezoidal).
3. Express x_i based on the chosen strategy.
4. Set up the summation: Σ(f(x_i) * Δx).
5. Take the limit as n approaches infinity.

Table for General Terms and Useful Conversions

Term	Description	Example
Riemann Sum	An approximation of the integral by summing the areas of rectangles under a curve.	For a simple function like f(x) = x² over [0, 1], a Riemann sum might involve summing the areas of rectangles under the curve.
Δx (Delta x)	The width of each subinterval in the partition of [a, b].	Δx = (b – a) / n, where n is the number of subintervals. For [0, 1] divided into 4 parts, Δx = 0.25.
n	The number of subintervals or rectangles used in the approximation.	Increasing n makes the approximation more accurate. For a simple approximation, n could be 4; for more accuracy, n could be 1000 or more.
Limit	The value that a function or sequence “approaches” as the input or index approaches some value.	lim_(n→∞) indicates the limit of an expression as n approaches infinity.

Example

Objective: Calculate the definite integral of the function f(x) = x^2 over the interval [0, 1] using the limit of a Riemann sum.

Step 1: Divide the interval [a, b], which is [0, 1], into n subintervals. Let’s choose n = 4 for simplicity, so Δx = (1 – 0) / 4 = 0.25.

Step 2: Choose a sampling strategy. We’ll use the right-endpoint for this example. Therefore, our sample points x_i for i = 1, 2, 3, 4 are 0.25, 0.5, 0.75, and 1, respectively.

Step 3: Calculate the areas of the rectangles. Each area is f(x_i) * Δx. So, we have:

• Rectangle 1: f(0.25) * Δx = (0.25)^2 * 0.25 = 0.015625
• Rectangle 2: f(0.5) * Δx = (0.5)^2 * 0.25 = 0.0625
• Rectangle 3: f(0.75) * Δx = (0.75)^2 * 0.25 = 0.140625
• Rectangle 4: f(1) * Δx = (1)^2 * 0.25 = 0.25

Step 4: Sum the areas to get the approximate integral: 0.015625 + 0.0625 + 0.140625 + 0.25 = 0.46875.

Step 5: Recognize that this is an approximation. To get the exact value of the definite integral, we take the limit as n approaches infinity. For simple functions like f(x) = x^2, we can use the formula integral from 0 to 1 of x^2 dx = [x^3/3] from 0 to 1 = (1/3) – (0/3) = 1/3.

Conclusion: Using a Riemann sum with 4 subintervals, we approximated the integral of x^2 over [0, 1] as 0.46875. As n increases, our approximation would get closer to the exact value, which calculus tells us is 1/3 or approximately 0.33333. This example illustrates how Riemann sums approximate the area under a curve, and how the limit of these sums as n approaches infinity gives us the definite integral of the function over the specified interval.

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