Understanding your integral result
The table below lists a few standard definite integrals with known exact values, useful for sanity-checking how this numerical calculator behaves on simple functions.
| Function f(x) | Interval [a, b] | Exact integral |
|---|---|---|
| x | [0, 1] | 0.5 |
| x² | [0, 1] | 0.33333333 (1/3) |
| 1/x | [1, e] | 1 |
| sin(x) | [0, pi] | 2 |
- This calculator returns a numeric approximation of the definite integral, not the antiderivative formula F(x) — indefinite integration (finding a general formula plus a constant C) is a different, symbolic operation this tool does not perform.
- Reversing the bounds negates the result: ∫ᵦᵃ f(x) dx = −∫ₐᵇ f(x) dx, a standard property of definite integrals, not an error condition.
- Integrating across a singularity or discontinuity within [a, b] — such as 1/x across x = 0, or tan(x) across x = π/2 — can produce an inaccurate result or none at all, because Simpson's rule assumes a continuous, well-behaved integrand across the interval.
- Accuracy is very high for smooth functions on a finite interval, but highly oscillatory or nearly singular integrands can be more difficult for any numerical method to converge on precisely.
What is a numerical definite integral?
A definite integral ∫ₐᵇ f(x) dx represents the net signed area between a function's graph and the x-axis over the interval from a to b: area where f(x) is positive counts positively, and area where f(x) is negative counts negatively. This calculator computes that number directly, without deriving a symbolic antiderivative F(x) — it never returns a formula like "x³/3 + C"; it returns only the single numeric value of the integral for the exact bounds entered.
The method used is adaptive composite Simpson's rule: the interval [a, b] is divided into many subintervals, a parabolic arc is fitted across each pair of subintervals, and the areas under these parabolas are summed. The number of subdivisions is doubled repeatedly until successive estimates agree to a tight numerical tolerance, which gives high accuracy for smooth, continuous functions.
This calculator also reports the average value of f over [a, b], defined by the mean value theorem for integrals as the integral divided by the width of the interval, (1 / (b − a)) × ∫ₐᵇ f(x) dx — the constant height a rectangle over [a, b] would need to have the same area as the integral.
How to use this integral calculator
- Enter a function of x, for example x^2, sin(x), or exp(-x^2). Multiplication must be written explicitly with *, and powers use ^.
- Enter the lower bound a and the upper bound b of the interval.
- If b is less than a, the calculator returns the negative of the integral from b to a — the standard sign convention for reversed bounds.
- Read the definite integral's value and, when a and b differ, the average value of the function over that interval.
The formula behind the numerical integral
Composite Simpson's rule approximates ∫ₐᵇ f(x) dx ≈ (h/3) × [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xₙ₋₁) + f(xₙ)], where the interval is divided into n equal subintervals of width h = (b − a)/n, and the coefficients alternate 4, 2, 4, 2, … between the endpoints. This calculator doubles n repeatedly and stops once successive estimates converge, ensuring high accuracy for smooth functions.
The average value of f on [a, b] follows from the mean value theorem for integrals: f_avg = (1 / (b − a)) × ∫ₐᵇ f(x) dx.
Worked example: f(x) = x², a = 0, b = 1. ∫₀¹ x² dx = 1/3 ≈ 0.33333333. Since the interval width b − a = 1, the average value equals the integral itself, 1/3.
Common mistakes
- Expecting a symbolic antiderivative such as "x³/3 + C" — this calculator returns only the numeric value of the definite integral for the exact bounds entered, not a general indefinite-integral formula.
- Integrating across a singularity or discontinuity, such as 1/x across x = 0 or tan(x) across x = π/2 — the numerical method assumes a continuous integrand and can produce an inaccurate or missing result.
- Treating a reversed sign (from swapping a and b) as an error — ∫ᵦᵃ f(x) dx = −∫ₐᵇ f(x) dx is a standard, expected property of definite integrals.
- Confusing the definite integral, a single number, with the indefinite integral (antiderivative), which is a family of functions differing by a constant C.
- Assuming a large numeric result always means a large positive area — the definite integral is a signed (net) area, so regions where f(x) is negative subtract from the total rather than adding to it.
Sıkça Sorulan Sorular
Does this calculator give me the antiderivative, like x³/3 + C?
No. This calculator computes only the numeric value of the definite integral for the specific bounds a and b you enter, using a numerical method. It does not perform symbolic integration and does not return a general antiderivative formula.
What numerical method does this integral calculator use?
Adaptive composite Simpson's rule, which approximates the area under the curve by fitting parabolic arcs across subintervals and summing their areas, refining the number of subintervals until the estimate converges to a tight numerical tolerance.
What happens if I enter the upper bound smaller than the lower bound?
The calculator returns the negative of the integral computed over the bounds in increasing order, following the standard convention that ∫ᵦᵃ f(x) dx = −∫ₐᵇ f(x) dx.
What does the average value result mean?
The average value of f on [a, b] is the integral divided by the width of the interval, (1 / (b − a)) × ∫ₐᵇ f(x) dx — the height of the rectangle over [a, b] that would have the same area as the region under the curve.
Can this calculator handle functions with a discontinuity in the interval?
Not reliably. Simpson's rule assumes the integrand is continuous and well-behaved across [a, b]. Integrating across a singularity, such as 1/x across x = 0, can produce an inaccurate result or no result at all.
What functions and operators are supported?
sin, cos, tan, asin, acos, atan, sqrt, cbrt, abs, ln (natural log), log (base-10 log), log2 (base-2 log), exp, floor, ceil, round, min, max; the constants pi, e and tau; the operators + − * / % ^; and the variable x, evaluated with trigonometric functions in radians.
Kaynaklar
- Stewart J. Calculus. Cengage Learning (the definite integral, Simpson's rule, and the mean value theorem for integrals).
- Burden RL, Faires JD. Numerical Analysis. Cengage Learning (composite Simpson's rule and its error bounds).
- Atkinson KE. An Introduction to Numerical Analysis. Wiley (numerical integration methods).
- NIST Digital Library of Mathematical Functions (DLMF), Chapter 3: Numerical Methods — quadrature. dlmf.nist.gov.