Quick Calculators

Numerical Integration Calculator: Trapezoidal & Simpson’s Rule

Numerical Integration Calculator

Numerical Integration Calculator

Approximate definite integrals using the Trapezoidal Rule and Simpson’s Rule.

Setup Integral

Sine Wave Parabola Bell Curve

Must be even for Simpson’s Rule.

Approximation Result

Step Size (Δx)
—
Calculated Area
—
True Value (Approx)
—
Error: –%

Visual Representation

Function Area

Numerical Integration Methods

Numerical integration algorithms allow us to calculate the definite integral $\int_a^b f(x) dx$ when an analytical antiderivative is difficult or impossible to find. These methods approximate the area under the curve using geometric shapes.

Trapezoidal Rule

Approximates the region under the graph of $f(x)$ as a trapezoid. It connects each point $(x_i, y_i)$ to $(x_{i+1}, y_{i+1})$ with a straight line.

Area ≈ (Δx/2) * [f(x₀) + 2f(x₁) + … + f(xₙ)]
Simpson’s Rule

Approximates the function using quadratic polynomials (parabolas). It requires an even number of subintervals ($n$). It is generally much more accurate than the Trapezoidal Rule.

Area ≈ (Δx/3) * [f(x₀) + 4f(x₁) + 2f(x₂) + … + f(xₙ)]

Error Analysis: The error in the Trapezoidal rule is proportional to $h^2$ (where $h = \Delta x$), while Simpson’s rule error is proportional to $h^4$. This means halving the step size in Simpson’s rule reduces the error by a factor of 16!

FAQ

Why must n be even for Simpson’s Rule?
Simpson’s rule works by fitting a parabola to groups of three points ($x_i, x_{i+1}, x_{i+2}$). This requires pairs of intervals. If you have an odd number of intervals, you cannot pair them all up perfectly.
What is the “True Value”?
In this tool, the “True Value” is calculated using a high-precision numerical integration (Simpson’s rule with 1000+ steps). It serves as a baseline to show you the error of your chosen approximation.

UNDERGRADUATE