List of mathematical formulas: “Approximate calculations of definite integrals

List of some basic formulas for approximate calculations of definite integrals:

1. Rectangle Method:

• ( I \approx \sum_{i=0}^{n-1} f(x_i) \Delta x )
• где ( \Delta x = \frac{b-a}{n} ), ( x_i = a + i \Delta x )

1. Trapezoid method:

• ( I \approx \frac{\Delta x}{2} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right) )

1. Simpson’s Method:

• ( I \approx \frac{\Delta x}{3} \left( f(a) + 4 \sum_{i=1, \text{ нечет.}}^{n-1} f(x_i) + 2 \sum_{i=2, \text{ чет.}}^{n-2} f(x_i) + f(b) \right) )

1. Method of average values:

• ( I \approx f(c) \cdot (b – a) )
• where ( c ) is a point in the interval ([a, b]).

1. Runge-Kutta method (for numerical integration):

• Used to solve differential equations, but can be adapted for integration.

These methods allow one to obtain approximate values ​​of definite integrals, and the choice of method depends on the required accuracy and the features of the function.