Many problems nowadays require approximations of high-dimensional integrals, because their analytical solutions cannot be provided. Among others, possible fields of application are Uncertainty Quantification and Machine Learning. Unfortunately high-dimensional problems suffer from the Curse of Dimensionality: Due to the exponential increase of the number of grid points the computation time grows rapidly. This makes calculations infeasible when the number of dimension grows. Possible solutions are Sparse Grids and the Sparse Grid Combination Technique, which are non-adaptive. However, many real-world applications have highly varying characteristics and thus require strategies that adapt to the given problem. A well-known approach is a dimension adaptive variant of the Combination Technique [GG03]. Another approach has been presented in [OB20]: a spatially adaptive variant with dimension-wise refinement. So far there has been little research towards high order methods for the above-mentioned approach. The goal of this thesis is to combine adaptive order methods and spatial adaptivity (with dimension-wise refinement). Therefore we investigate the well-known Romberg-Quadrature and generalize it for the application on adaptive grids. Thereafter, various variants of these theoretical results are incorporated into the sparseSpACE-framework and compared to adaptive as well as non-adaptive implementations of quadrature rules, such as Gauß- Legendre. The numerical results show that our adaptive extrapolation can significantly reduce the total number of distinct function evaluations to achieve a certain approximation tolerance threshold. This leads to shorter runtimes and a lower total number of refinements.