Numerical Integration

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Introduction

Midpoint Rule

Example

Trapezoid Rule

Example

Simpson's 1/3 Rule

Example

Summary of Common Quadrature Formulas

Name Formula Comments
Midpoint Rule \int_{a}^{b} f(x) \mathrm{d} x \approx (b-a) f\left(\tfrac{b+a}{2}\right)
  • Requires function values at interval midpoints f\left(\tfrac{b+a}{2}\right)
  • Requires equally spaced data.
Trapezoid Rule \int_{a}^{b} f(x) \mathrm{d} x \approx \tfrac{b-a}{2} \left[ f(b)+f(a) \right]
  • Can be applied to arbitrarily spaced data.
  • Convenient for tabulated data.
Simpson's 1/3 Rule \int_a^b f(x) \mathrm d x \approx \tfrac{\Delta x}{3} \left[ f(a) +4f\left(\tfrac{a+b}{2}\right) + f(b) \right]
  • Requires three equally spaced points on the interval [a,b]
  • On the interval [a,b], we have \Delta x = \tfrac{b-a}{2} and x_i = a + i \Delta x

Composite Rules: Quadrature

Algorithm

Examples

Matlab Tools for Quadrature