The Definite Integral Calculator evaluates an integral between two limits. Unlike an indefinite integral, a definite integral produces a value rather than a family of functions, and that value represents accumulated signed change over the interval.
Definite Integral Meaning
The standard notation is β«βα΅ f(x)dx. When f(x) is above the x-axis, the integral contributes positively; when it is below the axis, the contribution is negative. For a continuous function, the Fundamental Theorem of Calculus connects the integral to an antiderivative F: β«βα΅f(x)dx = F(b) β F(a).
Using the Calculator
Enter the function and the lower and upper limits. Check their order because reversing the limits changes the sign. If the function contains parentheses, powers, or trigonometric terms, enter them carefully so the intended expression is evaluated.
Example
For β«βΒ² xΒ² dx, an antiderivative is xΒ³/3. Evaluating the endpoints gives 8/3 β 0, so the definite integral is 8/3.
Signed Area vs Geometric Area
A definite integral measures signed area. If a curve crosses the x-axis and you need total geometric area, the interval may need to be split at the crossing points and the absolute areas combined.
Interpreting the Answer
The result depends on the function and interval, so it should not automatically be described as physical area. In applications it can represent accumulated distance, mass, probability, work, or another quantity depending on what the integrand means.