Welcome back. We've just completed the third and fourth modules introducing and discussing the differential calculus, which is really all about studying slopes of tangent lines to curves. This final module introduces and discusses integral calculus, which is really all about the study of areas under curves. Remarkably, slopes of tangent lines and areas under curves turn out to be intrinsically linked leading to the Fundamental Theorem of Calculus. This final module begins by illustrating how one can use areas under velocity curves to estimate displacement using averages of lower and upper rectangular approximations, and replicating thought experiments, originally due to the ancient Greeks involving limits of approximations to discover the formula for the area of a circle and the area under a parabola. We then formalize the method of Riemann sums using rectangular approximations to areas under curves over a given interval, leading to the definite integral, which is defined to be the limit of the Riemann sums, and captures precisely areas under a curve. We can calculate the definite integral exactly under certain conditions using the Fundamental Theorem of Calculus, providing a simple and elegant formula involving taking anti derivatives, which reverses the process of forming derivatives. We introduce indefinite integrals and illustrate the method of integration by substitution, closely related to the Chain Rule that you learned about in the previous module. We discuss properties of odd and even functions related to rotational and reflectional symmetry, and the logistic function, which modifies exponential growth by introducing an inhibition factor with important applications to population dynamics. Finally, we come full circle and provide you with some context and insight about how calculus was originally conceived by Newton in the 17th century by his remarkable estimate, the escape velocity of a rocket. Again, we hope that you'll find the material interesting and stimulating, that you find the videos helpful, and the practice and challenges provided by the many exercises beneficial. I look forward very much to your continued attention and participation.