Integral Calculator From A to B

This integral calculator computes the definite integral of a function from point a to point b. It provides both the numerical result and a visual representation of the area under the curve.

What is a Definite Integral?

A definite integral calculates the exact area under a curve between two points on the x-axis. It's represented as ∫[a,b] f(x) dx, where:

f(x) is the function being integrated
a is the lower limit of integration
b is the upper limit of integration

Definite integrals have many practical applications in physics, engineering, economics, and other fields where accumulation of quantities is important.

How to Use This Calculator

Enter the function you want to integrate in the function field (e.g., x^2, sin(x), etc.)
Specify the lower limit (a) and upper limit (b) of integration
Click "Calculate" to compute the integral
View the result and the visual representation of the area under the curve

Note: This calculator uses numerical integration methods for most functions. For exact results, you may need to use symbolic computation software.

The Integral Formula

The definite integral from a to b of a function f(x) is calculated using the following formula:

∫[a,b] f(x) dx = lim(n→∞) Σ[f(x_i) * Δx], where Δx = (b-a)/n

In practice, numerical methods like Simpson's rule or the trapezoidal rule are often used to approximate this sum.

Worked Examples

Example 1: Simple Polynomial

Calculate ∫[0,1] x^2 dx

The exact result is (x³)/3 evaluated from 0 to 1, which equals 1/3 ≈ 0.3333.

Example 2: Trigonometric Function

Calculate ∫[0,π] sin(x) dx

The exact result is -cos(x) evaluated from 0 to π, which equals 2.

Frequently Asked Questions

What types of functions can this calculator handle?: This calculator can handle most common mathematical functions including polynomials, trigonometric functions, exponentials, and logarithms.
Is the result always exact?: No, this calculator uses numerical approximation methods. For exact results, symbolic computation software is recommended.
Can I integrate functions with parameters?: Yes, you can integrate functions that include parameters, but you'll need to specify the parameter values in the function expression.
What if my function is undefined at one of the limits?: The calculator will attempt to compute the integral but may produce incorrect results if the function has singularities within the integration range.
How accurate are the results?: The accuracy depends on the numerical method used and the number of intervals. For most practical purposes, the results should be sufficiently accurate.