We investigate and implement various numerical algorithms for integration in exact real arithmetic based on domain theory. We cover three of the best known types of algorithms for integration. The method based on the construction of Darboux integral is the most direct but the least efficient case. We study two methods which belong to the class of closed Newton--Cotes formulae. These are the trapezoidal and Simpson's rules in their compound forms. Finally, we deal with the Gaussian quadrature technique, where our nodes are zeros of the Legendre polynomials. We tabulate the nodes and weights of the Gauss formulae, and give some of these values. We determine numerical integration methods which are efficient in the exact framework. Our algorithms can be used with any representation of exact reals; our implementation uses the framework of exact reals based on linear fractional transformations.
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