Enter a mathematical function, set your lower bound and upper bound, and this Definite Integral Calculator evaluates ∫f(x)dx numerically over your specified interval. You get the definite integral value along with a Riemann sum approximation and a visual breakdown of the area under the curve.
Definite Integral ∫f(x)dx
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Positive Area (above x-axis)
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Negative Area (below x-axis)
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Total Area |∫f(x)|dx
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Interval Width (b − a)
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Average Value of f(x)
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A definite integral ∫[a to b] f(x)dx represents the net signed area between the function f(x) and the x-axis over the interval [a, b]. Positive areas lie above the x-axis and negative areas lie below it. The result is a specific numerical value, unlike an indefinite integral which produces a family of functions.
This calculator uses numerical integration methods — Simpson's Rule, the Trapezoidal Rule, or the Midpoint Rule — to approximate the integral. The interval [a, b] is divided into n subintervals, and weighted sums of function values are computed. Simpson's Rule is generally the most accurate for smooth functions.
Definite integrals are used in physics (work, displacement, energy), engineering (signal processing, fluid mechanics), probability (computing probabilities from density functions), economics (consumer/producer surplus), and geometry (area, volume). Any time you need to accumulate a quantity over a continuous interval, a definite integral is the tool to use.
An indefinite integral ∫f(x)dx produces an antiderivative F(x) + C — a function, not a number. A definite integral ∫[a to b] f(x)dx applies the Fundamental Theorem of Calculus: it evaluates F(b) − F(a) to produce a specific numerical result. Bounds a and b are required for a definite integral.
You can enter standard math expressions using x as the variable. Use * for multiplication (e.g., 2*x), ^ for exponentiation (e.g., x^2), and built-in functions like sin(x), cos(x), tan(x), exp(x), log(x), sqrt(x), and abs(x). Always use parentheses to avoid ambiguity, e.g., 1/(x+1) instead of 1/x+1.
The definite integral computes the *net* signed area, meaning regions below the x-axis contribute negatively and can cancel out regions above. The total area is ∫|f(x)|dx, which treats all areas as positive. For example, ∫[−π to π] sin(x)dx = 0 (net), but the total area is 4.
Common errors include: forgetting the sign of regions below the x-axis, swapping the upper and lower bounds (which negates the result), using an incorrect antiderivative, and ignoring discontinuities within the interval. Always check that f(x) is defined and continuous (or integrable) over [a, b] before evaluating.
For most smooth, well-behaved functions, Simpson's Rule with n = 1000 subintervals gives accuracy to 6–10 significant digits — more than enough for practical purposes. Accuracy decreases near singularities or for highly oscillatory functions. Increasing n improves accuracy but also increases computation time.