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Numerical Integration Calculator

Trapezoidal and Simpson approximations with comparison to exact integrals

Parameters

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Show Trail

Controls

xⓘ

Calculated Values

Trapezoid:
0.34;0.34;
Simpson:
0.33;0.33;
Exact:
0.33;0.33;
Trap error %:
0.50;0.50;%
Simpson error %:
0.00;0.00;%

Examples

∫₀¹ x² dx

Exact value 1/3 ≈ 0.333; n = 10.

  • Exact: 0.330.33
  • Trapezoid: 0.340.34

∫₀^π sin(x) dx

Exact value 2.

  • Exact: 2.002.00
  • Simpson: 2.002.00

Visualization

Numerical Integration in Physics

Many physical quantities are defined as integrals: work W = ∫F·dx, charge Q = ∫I·dt, flux Φ = ∫B·dA, and areas under experimental curves. When f(x) has no elementary antiderivative, numerical methods approximate the definite integral.

The trapezoidal rule replaces the function on each subinterval by a straight line. With n equal panels of width h = (b−a)/n, ∫ₐᵇ f ≈ (h/2)[f₀ + 2f₁ + … + 2fₙ₋₁ + fₙ]. Global error scales as O(h²) for smooth functions.

Simpson’s rule fits a parabola through each pair of panels, giving weights 1, 4, 2, 4, …, 4, 1. It requires an even number of intervals and achieves O(h⁴) error — often far more accurate than trapezoid for the same n.

Increasing n refines the approximation but costs computation. In labs, noisy data may need smoothing before integration; propagate uncertainties with Error Propagation when f(xᵢ) come from measurements.

Numerical integration links to ODE solvers: integrating v(t) gives position — the same idea behind Projectile Motion simulations. Monte Carlo integration is an alternative for high-dimensional integrals.

Key Concepts

  • Panel width h = (b − a) / n
  • Trapezoid: chord approximation, error ~ O(h²)
  • Simpson: parabolic arcs, error ~ O(h⁴)
  • More panels → better accuracy, diminishing returns
  • Compare to exact value to validate code or method

Real-World Applications

  • Work from tabulated force vs displacement
  • Charge from current vs time graph
  • Center of mass from density distribution
  • Probability from Maxwell-Boltzmann distribution

Explore Further

More computational physics tools

  • Gradient Descent

    Iteratively minimize f(x) by following the negative gradient with animated path visualization.

  • 1D Heat Equation

    Finite-difference FTCS solution to the diffusion equation with animated temperature profiles.

  • 1D Wave Equation

    Leapfrog finite-difference solution to the wave equation with animated wave propagation.

  • ODE Solver

    Euler and Runge-Kutta 4 methods for first-order ODEs with comparison to analytic solutions.

  • Monte Carlo Intro

    Estimate π and integrals by random sampling — introduction to stochastic computational physics.

  • Newton-Raphson

    Solve nonlinear equations f(x) = 0 with tangent-line iterations — fast when the guess is good.

Physics Equations

Trapezoid:
∫abf≈h2[f0+2∑i=1n−1fi+fn]\int_a^b f \approx \frac{h}{2}\left[f_0 + 2\sum_{i=1}^{n-1}f_i + f_n\right]
Simpson:
∫abf≈h3[f0+4f1+2f2+⋯+fn]\int_a^b f \approx \frac{h}{3}\left[f_0 + 4f_1 + 2f_2 + \cdots + f_n\right]

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Define the integral and interval

Identify integrand f(x), limits a and b, and number of subintervals n.

Equation:

∫01x2 dx\int_{0}^{1} x^2 \, dx

Result:

n=10,h=(b−a)/n=0.100000n = 10, h = (b-a)/n = 0.100000

Explanation:

Smaller h improves accuracy but increases computation.

2

Step 2: Analytic value (if known)

Evaluate the definite integral exactly for comparison.

Equation:

[x33]01=13−033\left[\frac{x^3}{3}\right]_{0}^{1} = \frac{1^3 - 0^3}{3}

Result:

Iexact=0.333333I_{\text{exact}} = 0.333333
3

Step 3: Trapezoidal rule setup

Approximate area under f by trapezoids on each subinterval.

Equation:

Itrap≈h2[f0+2∑i=1n−1fi+fn]I_{\text{trap}} \approx \frac{h}{2}\left[f_0 + 2\sum_{i=1}^{n-1} f_i + f_n\right]

Explanation:

Sample f at x_i = a + ih for i = 0, 1, …, 10.

4

Step 4: Evaluate trapezoidal sum

f(a) = 0.0000, f(b) = 1.0000

Calculation:

Itrap≈0.335000I_{\text{trap}} \approx 0.335000

Result:

Trapezoid ≈ 0.335000

Explanation:

Error vs exact: 0.167%

5

Step 5: Simpson’s rule (even n)

Using 10 subintervals with weights 1, 4, 2, 4, …, 1.

Equation:

ISimp≈h3[f0+4f1+2f2+⋯+fn]I_{\text{Simp}} \approx \frac{h}{3}\left[f_0 + 4f_1 + 2f_2 + \cdots + f_n\right]

Calculation:

ISimp≈0.333333I_{\text{Simp}} \approx 0.333333

Result:

Simpson ≈ 0.333333

Explanation:

Error vs exact: 0.000% — usually smaller than trapezoid.

6

Step 6: Compare methods

Assess convergence; double n and see if result changes acceptably.

Result:

Exact=0.333333,Trap=0.335000,Simpson=0.333333Exact = 0.333333, Trap = 0.335000, Simpson = 0.333333

Frequently Asked Questions (FAQ)

Why is Simpson often more accurate?

It fits parabolas through triplets of points, capturing curvature better than straight-line trapezoids.

What if n is odd for Simpson?

This calculator uses the next even n automatically so Simpson’s formula applies.

When does trapezoid fail badly?

Sharp peaks, discontinuities, or very coarse n — increase n or use adaptive methods in research code.

Practice MCQs

  1. Doubling n in the trapezoid rule typically:
  2. Simpson’s rule requires:
  3. ∫₀¹ x² dx equals:
  4. Trapezoid approximates f on each panel by:
  5. For smooth functions, Simpson error order is:

Related Calculators

These tools connect to the same physics concepts used in this calculator.

Work and EnergyEnergy conservation checks after numerical integration of motion.Work W = ∫F·dx is often computed numerically from force data.Projectile MotionKinematic trajectories under gravity — compare simulated vs measured positions.Displacement integrals appear in trajectory analysis.Error PropagationCombine uncertainties when multiplying measured quantities.ODE SolverEuler and RK4 for equations of motion and decay.Monte Carlo IntroStochastic estimation of integrals and constants.Stochastic alternative when integrals are high-dimensional or irregular.Gradient DescentIteratively minimize f(x) by following the negative gradient with animated path visualization.1D Heat EquationFinite-difference FTCS solution to the diffusion equation with animated temperature profiles.1D Wave EquationLeapfrog finite-difference solution to the wave equation with animated wave propagation.