What is the fastest converging method which can be used to approximate definite integrals? What is the fastest-converging method which can be used to approximate definite integrals? I have come across various methods of finding definite integrals such as: finding anti derivatives, Simpson's rule, Trapezoidal rule, etc. But, which method converges the fastest?

Like @saulspatz said, there's no one answer to this. There are many interesting numerical integrator algorithms, some of which may be tailored towards specific types of functions. Matlab for instance uses Adaptive Quadrature. 
If you think about it, the Left-hand sum method is breaking the domain into segments and approximating each segment area with a degree 0 polynomial. For the trapezoid rule, a degree 1 polynomial. For Simpson's rule, degree 3. Boole's rule does it for degree 5. This can be generalized to whatever degree you want, which is what the Newton-Cotes formulae are.  
