Compute the definite integral of any function over [a, b] using Simpson's Rule with 1000 subintervals.
Enter a function and bounds a, b to compute the definite integral
This integral calculator computes the definite integral of any function over an interval [a,b] using Simpson's Rule with 1000 subintervals — a numerical integration technique that approximates the signed area under a curve without needing a symbolic antiderivative. Instead of applying integration rules like substitution or integration by parts to find a closed-form formula, it fits parabolas through thousands of tiny slices of your function and sums their areas. This makes it a fast way to evaluate a definite integral for coursework, check a hand-worked answer, or explore concepts like accumulated area, distance, or work directly from a function and two bounds.
It computes ∫[a,b] f(x) dx — the signed area between the curve y=f(x) and the x-axis from x=a to x=b — using Simpson's Rule with n=1000 subintervals. The result is a single number, not a symbolic antiderivative like x³/3; area above the x-axis is added and area below the x-axis is subtracted, and reversed bounds (a > b) are handled automatically with the correct sign.
Calculus students verifying a definite integral, physics and engineering students computing distance, work, or signal energy from a function, economics students estimating consumer or producer surplus, and anyone studying numerical methods who wants to see Simpson's Rule applied directly to real bounds. It's equally useful for double-checking an integral you solved symbolically by hand.
Many real-world functions don't have a simple closed-form antiderivative, and even functions that do can be evaluated numerically far faster than deriving and simplifying a symbolic formula. Numerical integration methods like Simpson's Rule are the same techniques used inside engineering software, physics simulations, and statistical tools whenever an exact analytic integral isn't practical to compute.
Physicists integrate a velocity function to find total distance traveled, or a force function to find work done. Engineers compute the area under a stress-strain curve or the energy in a signal. Economists integrate demand and supply curves to estimate consumer and producer surplus. Statisticians integrate a probability density function over an interval to find a probability.
How this numerical integration calculator turns a function and two bounds into a definite integral
Simpson's Rule fits a parabola through each pair of subintervals instead of a straight line, which follows curved functions far more closely than the Trapezoidal Rule for the same number of slices.
Using n=1000 subintervals makes the approximation extremely fine-grained, typically matching the exact analytic value to 6+ decimal places for smooth functions.
Area above the x-axis adds to the total; area below the x-axis subtracts. A definite integral can be zero or negative even when the curve is clearly visible on a graph.
From entering a function to verifying your result
Type an expression in x using + − * / ^, parentheses, and supported functions like sin(x), cos(x), sqrt(x), exp(x), ln(x), log(x), and abs(x) — for example x^2.
Type the numeric value where the interval of integration begins.
Type the numeric value where the interval of integration ends. If b is smaller than a, the calculator handles the sign automatically.
The calculator applies Simpson's Rule with 1000 subintervals to approximate ∫[a,b] f(x) dx.
Check the computed integral value plus a note on the method used, including any sign flip if bounds were reversed.
For simple functions like x², compare the numerical result to the exact antiderivative evaluated at the bounds.
∫[0,3] x² dx — a function with a known, easily verified exact integral
Find the definite integral of f(x) = x² from a = 0 to b = 3, and compare it against the exact antiderivative x³/3.
Explanation: Because x² has a simple, well-known exact antiderivative (x³/3), this example makes it easy to see that Simpson's Rule's numerical output (9.000000) matches the true calculus answer (9) almost exactly. For functions without a simple closed-form antiderivative, this same process still works — you just won't have an independent exact answer to compare against, which is where trusting the high subinterval count (n=1000) becomes especially useful.
What the numbers in your result actually represent
| Output | What It Means | Example Reading |
|---|---|---|
| Integral value | The signed area between the curve and the x-axis from a to b | ∫[0,3] x² dx = 9 → the area under the parabola from x=0 to 3 is 9 square units |
| Negative result | The region below the x-axis outweighs any region above it over [a,b] | ∫f(x)dx = −5 means the net signed area is negative |
| Sign flip when a > b | The calculator integrates over [b,a] then negates the result | ∫[3,0] f(x) dx = −∫[0,3] f(x) dx |
| Error / undefined result | The function could not be evaluated somewhere in [a,b] — often a discontinuity or domain restriction | 1/x from −1 to 1 fails because x=0 is outside the domain |
Manual verification: for polynomials and other functions with a known antiderivative F(x), use the Fundamental Theorem of Calculus — compute F(b) − F(a) by hand and compare it to the calculator's result. A close match (typically 5-6 decimal places) confirms the numerical estimate is trustworthy. For functions without a simple closed form, the high subinterval count (n=1000) is generally sufficient evidence of accuracy on its own.
A definite integral returns signed area, so a region above the x-axis can be cancelled exactly by an equally sized region below it. For example, sin(x) integrated from 0 to 2π returns 0 because the two halves balance. If you need the plain geometric area instead, integrate abs(sin(x)) over the same bounds.
The theoretical error bound for Simpson's Rule depends on the fourth derivative of f(x) and the subinterval width h, but this tool fixes n at 1000 so you cannot change h directly. A practical check is to compare the result with the exact antiderivative for a function whose closed form you know, or to split the interval in half, integrate each piece, and add them — if the total matches the single run, the approximation has very likely converged.
The curve looks symmetric, but f(x) = 1/x grows without bound as x approaches 0 from either side, so the integral is improper and diverges. The calculator cannot evaluate the sample point at x = 0 either, so it reports an error rather than returning the symmetric-looking zero that a careless glance at the graph might suggest.
No — this tool only evaluates proper definite integrals over a finite interval [a, b], so you cannot type infinity as a bound. To approximate an improper integral to infinity, choose a very large finite upper bound instead; many integrands decay fast enough that a bound like 10 already captures the value to several decimal places (for example e^(-x^2) from 0 to 10 approaches √π/2).
If your function has a simple antiderivative, the theorem gives an exact answer by evaluating F(b) − F(a), which matters when an assignment requires a symbolic result. Numerical integration is the better tool when no closed-form antiderivative exists, when only a decimal answer is needed, or when you want a fast cross-check of a hand-worked integral.
Integration in calculus assumes angles measured in radians, and sin(x), cos(x), and tan(x) follow that convention here. If your problem is stated in degrees, convert the bounds — an integral from 0° to 180° becomes 0 to pi, since 180° equals π radians. For a degree-based argument inside the function, multiply it by pi/180 in the expression you type.
Where a fast, numerical definite integral is genuinely useful
Verify a hand-worked definite integral before submitting homework.
Quickly check integration practice problems for entrance exams and standardized tests.
Integrate a velocity function to find total distance traveled over a time interval.
Integrate a variable force function to compute the work done moving an object.
Find the area under a stress-strain curve or the energy contained in a signal.
Estimate surplus as the area between supply, demand, and price curves.
Integrate a probability density function over an interval to find a probability.
Estimate area-under-curve style metrics used in evaluating models and distributions.
Compute area and volume quantities used in rendering and physical simulations.
Find the area of irregular, curve-bounded shapes defined by a function.
Accumulate a measured rate (like flow or emissions) over time into a total quantity.
Show students how Simpson's Rule approximates area with parabolic segments.
A fast sanity check for symbolic integration results over specific bounds.
What this numerical integral calculator does well, and where it can't replace a computer algebra system
Three ways to numerically approximate a definite integral
| Method | Approach | Typical Accuracy |
|---|---|---|
| Trapezoidal Rule | Approximates the area under each subinterval with a straight-line trapezoid | O(h²) — good for smooth, well-behaved functions |
| Simpson's Rule (used here) | Fits a parabola through each pair of subintervals | O(h⁴) — typically much more accurate than trapezoidal for the same n |
| Monte Carlo integration | Randomly samples points and averages function values across the interval | Converges slowly (~O(1/√N)) but scales well to higher-dimensional integrals |
Summary: This integral calculator gives you an instant, numerically accurate definite integral using Simpson's Rule with 1000 subintervals. Pair it with the Derivative Calculator and Limit Calculator to cover differentiation, integration, and limits — the three foundations of first-semester calculus — all computed numerically in your browser.
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