We have chosen suitable multi-variable probability density functions and obtained corresponding random variates using the random sampling method called acceptance-rejection method. This book provides hands-on and practical demonstrations of evaluation of multi-dimensional or high-dimensional definite integrals using Monte Carlo method. We have evaluated 2, 3, 5, 7 and 10 dimensional definite integrals. We performed symbolic computations using programs written in Mathematica(R), in which we performed the variate sampling and evaluated the integrals. The book is self-contained.