Numerical integration approximates definite integrals using weighted sums of function values at discretized points. Common integration rules include the rectangular rule, which uses rectangles of width Δx; the trapezoidal rule, which uses trapezoids; and Simpson's rule, which uses a quadratic polynomial to achieve higher accuracy. The document provides examples applying these rules to calculate the integral of f(x)=x^3 from 1 to 2, demonstrating that Simpson's rule provides a perfect estimation while the other rules have some error.