Master calculus with precision using our Integral Calculator. This tool provides instant solutions for definite integrals, allowing you to compute the area under a curve, total accumulation, and net change for any continuous function across a specified interval.
Integral Calculator
Integral Calculator Formula
Where F(x) is the antiderivative of f(x).
Formula Source: Khan Academy – Integral Calculus
Variables:
- f(x): The integrand function you wish to integrate.
- a: The lower bound of the integration interval.
- b: The upper bound of the integration interval.
- dx: The differential representing an infinitesimal change in x.
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What is an Integral Calculator?
An integral calculator is a specialized mathematical tool used to compute the integral of a function. In calculus, integration is the reverse process of differentiation and is fundamental for calculating areas, volumes, and central points.
The definite integral computes the net area between the function’s graph and the x-axis from point A to point B. It is essential in physics for determining work, distance, and probability density in statistics.
How to Calculate a Definite Integral (Example)
To find the integral of $f(x) = 2x$ from $x=1$ to $x=3$:
- Identify the function: $f(x) = 2x$.
- Find the antiderivative: $F(x) = x^2$.
- Evaluate at the upper limit: $F(3) = 3^2 = 9$.
- Evaluate at the lower limit: $F(1) = 1^2 = 1$.
- Subtract lower from upper: $9 – 1 = 8$. The result is 8.
Frequently Asked Questions (FAQ)
What is the difference between definite and indefinite integrals?
A definite integral has limits (a and b) and results in a number, while an indefinite integral represents a family of functions (antiderivatives) plus a constant C.
Can the integral result be negative?
Yes. If the function lies below the x-axis, the definite integral value will be negative, representing a “negative area” relative to the axis.
How accurate is this numerical integration?
This calculator uses Simpson’s Rule with 10,000 intervals, providing high precision for most smooth, continuous functions.
What happens if the function is discontinuous?
Numerical integration may produce incorrect results if there is a vertical asymptote or jump discontinuity within the limits [a, b].