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Numerical Integration Calculator

Compare Simpson's rule, trapezoidal rule, and midpoint rule for numerical integration with method comparison.

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Basic Polynomial

Integrate x^2 from 0 to 1 using Simpson's rule. Exact answer: 1/3.

Key values: x^2 · [0, 1] · Simpson's rule

Trigonometric Integral

Integrate sin(x) from 0 to pi. Exact answer: 2.

Key values: sin(x) · [0, pi] · Exact: 2

Area Under a Bell Curve

Approximate the Gaussian integral e^(-x^2) from -3 to 3.

Key values: e^(-x^2) · [-3, 3] · Gaussian

Documentation

Why Numerical Integration?

Many integrals have no closed-form antiderivative. Classic examples include:

\int e^{-x^2} dx \qquad \int \frac{\sin x}{x} \, dx \qquad \int \sqrt{1 + x^3} \, dx

For these functions, and for data given as discrete points (e.g., experimental measurements), numerical methods are the only option.

Method Comparison

Method	Error order	Function evals	Best for
Left/Right Riemann	$O(h)$	n	Conceptual understanding
Midpoint rule	$O(h^2)$	n	Quick estimates
Trapezoidal rule	$O(h^2)$	n + 1	Equally spaced data
Simpson's rule	$O(h^4)$	n + 1 (n even)	Smooth functions
Gaussian quadrature	$O(h^{2n})$	n	Maximum accuracy per eval

Here $h = \Delta x = (b-a)/n$ is the step size. Doubling $n$ (halving $h$ ) cuts Simpson's error by a factor of 16, vs. only 4 for the trapezoidal rule.

Error Bounds

The error in each method depends on the function's higher derivatives:

|E_T| \leq \frac{(b-a)^3}{12n^2} \max |f''(x)| \quad \text{(Trapezoidal)}

|E_S| \leq \frac{(b-a)^5}{180n^4} \max |f^{(4)}(x)| \quad \text{(Simpson's)}

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.

Adaptive Quadrature

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:

Estimate the integral with $n$ intervals
Re-estimate with $2n$ intervals
If the estimates agree within tolerance, accept
Otherwise, subdivide and recurse on each half

This is how production numerical libraries (SciPy, MATLAB) integrate functions efficiently.

Frequently Asked Questions

When do I need numerical integration?

Numerical integration is necessary when the integrand has no closed-form antiderivative (e.g., $e^{-x^2}$ , $\frac{\sin x}{x}$ , $\sqrt{1 + x^3}$ ), when you have discrete data points rather than a formula, or when the antiderivative is too complex to evaluate by hand.

Which numerical integration method should I use?

Simpson's rule is the best default for smooth functions due to its $O(h^4)$ convergence. Use the trapezoidal rule for equally spaced experimental data or periodic functions. Use the midpoint rule when the function has endpoint singularities.

What is the difference between $O(h^2)$ and $O(h^4)$ convergence?

$O(h^2)$ means halving the step size reduces the error by a factor of 4. $O(h^4)$ means halving the step size reduces the error by a factor of 16. Simpson's rule ( $O(h^4)$ ) converges much faster than trapezoidal or midpoint rules ( $O(h^2)$ ).

What is adaptive quadrature?

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.

Can Simpson's rule integrate polynomials exactly?

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.

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