This work aims to address green's theorem and applications. Of a bibliographic character, it emphasizes the applications of area calculations and Moments of Inertia, establishing for their results, definitions and theorems, guiding concepts of great importance, such as: Vector and Integral Fields of Line. Green's Theorem interfaces the line integral to the logo of a certain partially smooth single closed curve with the double integral over the region bounded by this curve, expressed by∮ 𝑀(𝑥,𝑦)𝑑𝑥 + 𝑁(𝑥,𝑦)𝑑𝑦 = ∬ − 𝑑𝐴 . This theorem, which is an important Vector Differential Calculus Theorem, makes you have the option to choose to work single integral instead of double integral on a region, and vice versa, thus realizing green's theorem as a tool important to collaborate in solving difficult-to-solve problems. With this context, it discusses the importance of showing its application in the area calculation of regions determined by simple and closed curves and also the relationship of the Green Theorem and the Moment of Inertia. For calculation of area we adopt the theorem which is a consequence of green's theorem, expressed by ∮ 𝑥𝑑𝑦 − 𝑦𝑑𝑥 = 𝐴 .