This review explains the limitations of the Riemann integral and focuses on the more general concept. It has been suggested that a better route is to abandon the Riemann integral for Lebesgue integral and explained the importance of Lebesgue integral in real analysis and other fields of mathematical science. Keywords: Riemann integral; Lebesgue integral; Fourier series DOI: 10. https://intcariuma.shop/product-category/accessories/