In this lesson we are going to go through the logical framework that defines a definite integral. The main goal of this lesson is to show that when $\lim_{||P\,||\to 0}\,L(P)$ and $\lim_{||P\,||\to 0}\,U(P)$ are equal, then they are equal to $\int_a^bf(x)\,dx.$ We also provide three equivalent definitions of a definite integral.