Numerical Integration Made Easy
Calculate definite integrals using Simpson's Rule, a numerical method that provides more accurate approximations than the trapezoidal rule. Perfect for integrals that are difficult or impossible to solve analytically.
Welcome to our Simpson's Rule Calculator! Use this tool to approximate the definite integral of a function over a specified interval using Simpson's Rule, a powerful method in numerical integration.
Simpson's Rule is particularly effective for polynomials and smooth functions, providing a more accurate approximation compared to other numerical methods like the Trapezoidal Rule.
Simpson's Rule is a method for numerical integration, the numerical approximation of definite integrals. It provides a way to estimate the area under a curve by dividing the area into parabolic segments, offering greater accuracy than methods like the Trapezoidal Rule.
For an even number of intervals , Simpson's Rule is given by:
Where , and are the lower and upper limits of integration, respectively. Simpson's Rule assumes that the function can be approximated by a second-degree polynomial within each subinterval.
To further enhance your mathematical computations, explore our other calculators in the Mathematics category:
These tools are designed to assist you with a wide range of mathematical problems, ensuring accurate and efficient solutions every time.
Simpson's Rule approximates the area under a curve by fitting parabolas through three consecutive points. The formula: ∫[a,b] f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)], where h = (b-a)/n and n is even. This method typically gives better accuracy than rectangular or trapezoidal approximations.
Use Simpson's Rule when: the integrand has no elementary antiderivative, you need numerical approximation, the function is continuous on the interval, you want better accuracy than trapezoidal rule, and computational resources allow. Common in engineering, physics, and scientific computing.
Accuracy improves with more subintervals (larger n). Error is proportional to h⁴, making it more accurate than methods with h² error. For smooth functions, Simpson's Rule often gives excellent results with relatively few subintervals. Always verify results by increasing n and comparing.
