This pivotal chapter introduces the Lebesgue integral for bounded functions, drawing parallels with the Riemann integral to help the reader understand that the Lebesgue integral generalizes the Riemann integral. The definition is extended to unbounded functions as well. Measurability of the function is not assumed initially, but it is proved that if f is Lebesgue integrable, then f is measurable. Other definition of the Lebesgue integral is also included, and their equivalence is established.The Riemann integrability of a bounded derivative and the Volterra function are discussed. Properties of the Lebesgue integral, including convergence theorems, are studied. Additionally, the chapter compares the Improper Riemann integral, the Newton integral, and the Lebesgue integral, highlighting their independence and unique characteristics.