Limit Definition of A Definite Integral Calculator

What is the Limit Definition of a Definite Integral?

The limit definition of a definite integral is a fundamental concept in calculus that connects the idea of area under a curve to the concept of limits. It provides a rigorous foundation for understanding how integrals are calculated.

At its core, the definition states that the definite integral of a function f(x) from a to b is the limit of a Riemann sum as the number of subintervals approaches infinity. This means we're essentially calculating the area under the curve by dividing it into an infinite number of infinitely thin rectangles.

How to Calculate Using the Limit Definition

Calculating a definite integral using its limit definition involves several steps:

1. Divide the interval [a, b] into n equal subintervals of width Δx = (b - a)/n
2. Choose sample points xi in each subinterval
3. Calculate the height of each rectangle as f(xi)
4. Sum the areas of all rectangles: Σ f(xi)Δx
5. Take the limit as n approaches infinity

In practice, we use the limit definition to derive the Fundamental Theorem of Calculus, which provides a much simpler method for evaluating definite integrals.

Formula

This formula shows that the definite integral is the limit of a sum of areas of rectangles under the curve as the number of rectangles approaches infinity.

Worked Example

Let's calculate the definite integral of f(x) = x² from 0 to 1 using the limit definition.

1. Divide [0,1] into n subintervals: Δx = 1/n
2. Choose right endpoints: xi = i/n
3. Sum the areas: Σ (i/n)² (1/n) = (1/n³) Σ i²
4. We know Σ i² = n(n+1)(2n+1)/6
5. So the sum becomes (1/n³) * [n(n+1)(2n+1)/6] = (n+1)(2n+1)/6n²
6. Take the limit as n→∞: lim (n+1)(2n+1)/6n² = 1/3

The result is 1/3, which matches the known integral of x² from 0 to 1.