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Integrate x^2 from 0 to 1 using Simpson's rule. Exact answer: 1/3.
Key values: x^2 · [0, 1] · Simpson's rule
Integrate sin(x) from 0 to pi. Exact answer: 2.
Key values: sin(x) · [0, pi] · Exact: 2
Approximate the Gaussian integral e^(-x^2) from -3 to 3.
Key values: e^(-x^2) · [-3, 3] · Gaussian
Enter a function f(x) with bounds and click Calculate to see the numerical integral, error estimate, and interactive Riemann sum visualization.
Many integrals have no closed-form antiderivative. Classic examples include:
For these functions, and for data given as discrete points (e.g., experimental measurements), numerical methods are the only option.
| Method | Error order | Function evals | Best for |
|---|---|---|---|
| Left/Right Riemann | n | Conceptual understanding | |
| Midpoint rule | n | Quick estimates | |
| Trapezoidal rule | n + 1 | Equally spaced data | |
| Simpson's rule | n + 1 (n even) | Smooth functions | |
| Gaussian quadrature | n | Maximum accuracy per eval |
Here is the step size. Doubling (halving ) cuts Simpson's error by a factor of 16, vs. only 4 for the trapezoidal rule.
The error in each method depends on the function's higher derivatives:
Key insight: Simpson's rule is exact for polynomials of degree ≤ 3, even though it only uses parabolic approximations. This is because the error term depends on the 4th derivative, which is zero for cubics.
Fixed-step methods waste evaluations on smooth regions and may miss sharp features. Adaptive quadrature automatically refines the step size where the function changes rapidly:
This is how production numerical libraries (SciPy, MATLAB) integrate functions efficiently.
Numerical integration is necessary when the integrand has no closed-form antiderivative (e.g., , , ), when you have discrete data points rather than a formula, or when the antiderivative is too complex to evaluate by hand.
Simpson's rule is the best default for smooth functions due to its convergence. Use the trapezoidal rule for equally spaced experimental data or periodic functions. Use the midpoint rule when the function has endpoint singularities.
means halving the step size reduces the error by a factor of 4. means halving the step size reduces the error by a factor of 16. Simpson's rule () converges much faster than trapezoidal or midpoint rules ().
Adaptive quadrature automatically refines the step size in regions where the function changes rapidly while using fewer evaluations where the function is smooth. This is the approach used by production libraries like SciPy and MATLAB.
Yes, Simpson's rule is exact for polynomials of degree 3 or less, even though it uses parabolic (degree 2) approximations. This is because the error bound depends on the fourth derivative, which is zero for cubics.