List of some basic formulas for approximate calculations of definite integrals:
- 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 )
- Trapezoid method:
- ( I \approx \frac{\Delta x}{2} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right) )
- 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) )
- Method of average values:
- ( I \approx f(c) \cdot (b – a) )
- where ( c ) is a point in the interval ([a, b]).
- 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.