Numerical methods can be used to approximate definite integrals when the integral cannot be evaluated analytically or when the algebraic evaluation would be too complex. There are several numerical methods available for approximating definite integrals, and I will highlight a few of the commonly used ones.

  1. Riemann sum: This method approximates the definite integral by dividing the interval of integration into several subintervals and approximating the area under the curve with rectangles. The approximation improves as the number of subintervals increases, and it approaches the exact value of the integral as the width of each subinterval approaches zero. Riemann sums come in different forms such as left, right, and midpoint sums, depending on the location of the rectangles.

  2. Trapezoidal rule: This method approximates the integral by replacing the curve with a series of trapezoids. The area of each trapezoid is calculated, and the sum of these areas gives an estimate of the integral. Similar to Riemann sums, the accuracy of the approximation increases as the number of trapezoids increases.

  3. Simpson's rule: This method approximates the integral using quadratic polynomials to model the curve. The integration interval is divided into subintervals, and on each subinterval, a quadratic polynomial is constructed using three points. The area under each quadratic polynomial is calculated, and their sum gives an approximation of the integral. Simpson's rule is generally more accurate than the trapezoidal rule and requires fewer subintervals for the same level of accuracy.

  4. Gaussian quadrature: This method approximates the integral using weighted sums of function values at specific points. It utilizes a set of predefined nodes and weights that are optimized for accuracy. Gaussian quadrature can achieve high accuracy, especially when the integrand is smooth.

  These methods allow us to numerically estimate definite integrals by breaking them down into simpler, more manageable calculations. By using a large number of subintervals or quadrature points, we can improve the accuracy of the approximation. However, it is important to note that numerical methods are still approximations and may introduce some level of error.